From pas at lists.imstat.org Fri Jan 4 05:27:38 2008 From: pas at lists.imstat.org (Probability Abstract Service) Date: Fri, 4 Jan 2008 12:27:38 +0100 Subject: [PAS] Probability Abstracts 101 Message-ID: Probability Abstracts 101 This document contains abstracts 6228-6510 from November-1-2007 to December-31-2007. They have been mailed on January 4th, 2008. This letter can be also found on line at http://pas.imstat.org/Letters/letter_101.shtml --------------------------------------------------------------- 6228. ON FRACTIONAL BROWNIAN MOTION LIMITS IN ONE DIMENSIONAL NEAREST-NEIGHBOR SYMMETRIC SIMPLE EXCLUSION Magda Peligrad and Sunder Sethuraman A well-known result with respect to the one dimensional nearest- neighbor symmetric simple exclusion process is the convergence to fractional Brownian motion with Hurst parameter 1/4, in the sense of finite-dimensional distributions, of the subdiffusively rescaled current across the origin, and the subdiffusively rescaled tagged particle position. The purpose of this note is to improve this convergence to a functional central limit theorem, with respect to the uniform topology, and so complete the solution to a conjecture in the literature with respect to simple exclusion processes. http://arxiv.org/abs/0711.0017 --------------------------------------------------------------- 6229. THE QUENCHED CRITICAL POINT OF A DILUTED DISORDERED POLYMER MODEL Erwin Bolthausen and Francesco Caravenna and B\'eatrice de Tili \`ere We consider a model for a polymer interacting with an attractive wall through a random sequence of charges. We focus on the so-called diluted limit, when the charges are very rare but have strong intensity. In this regime, we determine the quenched critical point of the model, showing that it is different from the annealed one. The proof is based on a rigorous renormalization procedure. Applications of our results to the problem of a copolymer near a selective interface are discussed. http://arxiv.org/abs/0711.0141 --------------------------------------------------------------- 6230. ISOPERIMETRY AND ROUGH PATH REGULARITY Peter Friz and Harald Oberhauser Optimal sample path properties of stochastic processes often involve generalized H\"{o}lder- or variation norms. Following a classical result of Taylor, the exact variation of Brownian motion is measured in terms of $\psi (x) \equiv $ $x^{2}/\log \log (1/x) $ near $0+$. Such $\psi $- variation results extend to classes of processes with values in abstract metric spaces. (No Gaussian or Markovian properties are assumed.) To establish integrability properties of the $\psi $-variation we turn to a large class of Gaussian rough paths (e.g. Brownian motion and L\'{e}vy's area viewed as a process in a Lie group) and prove Gaussian integrability properties using Borell's inequality on abstract Wiener spaces. The interest in such results is that they are compatible with rough path theory and yield certain sharp regularity and integrability properties (for iterated Stratonovich integrals, for example) which would be difficult to obtain otherwise. At last, $\psi $- variation is identified as robust regularity property of solutions to (random) rough differential equations beyond semimartingales. http://arxiv.org/abs/0711.0163 --------------------------------------------------------------- 6231. ENTROPIC PROJECTIONS AND DOMINATING POINTS Christian L\'eonard (MODAL'x and Cmap) Generalized entropic projections and dominating points are solutions to convex minimization problems related to conditional laws of large numbers. They appear in many areas of applied mathematics such as statistical physics, information theory, mathematical statistics, ill-posed inverse problems or large deviation theory. By means of convex conjugate duality and functional analysis, criteria are derived for their existence. Representations of the generalized entropic projections are obtained: they are the ``measure component" of some extended entropy minimization problem. http://arxiv.org/abs/0711.0206 --------------------------------------------------------------- 6232. KERNEL CONVERGENCE ESTIMATES FOR DIFFUSIONS WITH CONTINUOUS COEFFICIENTS Claudio Albanese We are interested in the kernel of one-dimensional diffusion equations with continuous coefficients as evaluated by means of explicit discretization schemes of uniform step $h>0$ in the limit as $h\to0$. We consider both semidiscrete triangulations with continuous time and explicit Euler schemes with time step small enough for the method to be stable. We find sharp uniform bounds for the convergence rate as a function of the degree of smoothness which we conjecture. The bounds also apply to the time derivative of the kernel and its first two space derivatives. Our proof is constructive and is based on a new technique of path conditioning for Markov chains and a renormalization group argument. Convergence rates depend on the degree of smoothness and H\"older differentiability of the coefficients. We find that the fastest convergence rate is of order $O(h^2)$ and is achieved if the coefficients have a bounded second derivative. Otherwise, explicit schemes still converge for any degree of H\"older differentiability except that the convergence rate is slower. H\"older continuity itself is not strictly necessary and can be relaxed by an hypothesis of uniform continuity. http://arxiv.org/abs/0711.0132 --------------------------------------------------------------- 6233. LINEAGE-THROUGH-TIME PLOTS OF BIRTH-DEATH PROCESSES Tanja Gernhard and Dennis Wong We calculate the density and expectation for the number of lineages in a reconstructed tree with $n$ extant species. This is done with conditioning on the age of the tree as well as with assuming a uniform prior for the age of the tree. http://arxiv.org/abs/0711.0269 --------------------------------------------------------------- 6234. CONTINUUM PERCOLATION AT AND ABOVE THE UNIQUENESS TRESHOLD ON HOMOGENEOUS SPACES Johan Tykesson We consider the Poisson Boolean model of continuum percolation on a homogeneous Riemannian manifold $M$. Let $lambda$ be intensity of the Poisson process in the model and let $lambda_u$ be the infimum of the set of intensities that a.s. produce a unique unbounded component. We show that above $\lambda_u$ there is a.s. a unique unbounded component. We also study what happens at $\lambda_u$ for some spaces. In particular, if $M$ is the product of the hyperbolic disc and the real line, then at $\lambda_u$ there is a.s. not a unique unbounded component. The results are inspired by results for Bernoulli bond percolation on graphs due to Haggstrom, Peres and Schonmann. http://arxiv.org/abs/0711.0307 --------------------------------------------------------------- 6235. INTERMITTENT ESTIMATION OF STATIONARY TIME SERIES G. Morvai and B. Weiss Let $\{X_n\}_{n=0}^{\infty}$ be a stationary real-valued time series with unknown distribution. Our goal is to estimate the conditional expectation of $X_{n+1}$ based on the observations $X_i$, $0\le i\le n$ in a strongly consistent way. Bailey and Ryabko proved that this is not possible even for ergodic binary time series if one estimates at all values of $n$. We propose a very simple algorithm which will make prediction infinitely often at carefully selected stopping times chosen by our rule. We show that under certain conditions our procedure is strongly (pointwise) consistent, and $L_2$ consistent without any condition. An upper bound on the growth of the stopping times is also presented in this paper. http://arxiv.org/abs/0711.0350 --------------------------------------------------------------- 6236. NONPARAMETRIC INFERENCE FOR ERGODIC, STATIONARY TIME SERIES G. Morvai and S. Yakowitz and and L. Gyorfi The setting is a stationary, ergodic time series. The challenge is to construct a sequence of functions, each based on only finite segments of the past, which together provide a strongly consistent estimator for the conditional probability of the next observation, given the infinite past. Ornstein gave such a construction for the case that the values are from a finite set, and recently Algoet extended the scheme to time series with coordinates in a Polish space. The present study relates a different solution to the challenge. The algorithm is simple and its verification is fairly transparent. Some extensions to regression, pattern recognition, and on-line forecasting are mentioned. http://arxiv.org/abs/0711.0367 --------------------------------------------------------------- 6237. PERIOD LENGTHS FOR ITERATED FUNCTIONS Eric Schmutz For random maps, the expected value of the order (i.e. the period of the sequence of compositional iterates) is approximated asymptotically. It is much smaller than the expected value for the product of the cycle lengths. http://arxiv.org/abs/0711.0312 --------------------------------------------------------------- 6238. PREDICTION FOR DISCRETE TIME SERIES G. Morvai and B. Weiss Let $\{X_n\}$ be a stationary and ergodic time series taking values from a finite or countably infinite set ${\cal X}$. Assume that the distribution of the process is otherwise unknown. We propose a sequence of stopping times $\lambda_n$ along which we will be able to estimate the conditional probability $P(X_{\lambda_n+1}=x|X_0,...,X_{\lambda_n})$ from data segment $(X_0,...,X_{\lambda_n})$ in a pointwise consistent way for a restricted class of stationary and ergodic finite or countably infinite alphabet time series which includes among others all stationary and ergodic finitarily Markovian processes. If the stationary and ergodic process turns out to be finitarily Markovian (among others, all stationary and ergodic Markov chains are included in this class) then $ \lim_{n\to \infty} {n\over \lambda_n}>0$ almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then $\lambda_n$ is upperbounded by a polynomial, eventually almost surely. http://arxiv.org/abs/0711.0471 --------------------------------------------------------------- 6239. ORDER ESTIMATION OF MARKOV CHAINS G. Morvai and B. Weiss We describe estimators $\chi_n(X_0,X_1,...,X_n)$, which when applied to an unknown stationary process taking values from a countable alphabet $ {\cal X}$, converge almost surely to $k$ in case the process is a $k$-th order Markov chain and to infinity otherwise. http://arxiv.org/abs/0711.0472 --------------------------------------------------------------- 6240. MARKOV PROCESSES WITH PRODUCT-FORM STATIONARY DISTRIBUTION Krzysztof Burdzy and David White We study a class of Markov processes with finite state space and continuous time that have product form stationary distributions. We obtain a number of examples that can generate conjectures for diffusions with inert drift. http://arxiv.org/abs/0711.0493 --------------------------------------------------------------- 6241. AN ALTERNATIVE CONSTRUCTION OF THE STRONG EMBEDDING FOR THE SIMPLE RANDOM WALK Sourav Chatterjee We give a new proof of the Komlos-Major-Tusnady embedding theorem for the simple random walk. The only external tool that we use is the Schauder-Tychonoff fixed point theorem for locally convex spaces. Besides that, the proof is almost entirely based on a series of soft arguments and easy inequalities, and no hard computations (implicit or explicit) are involved. This provides the first genuine alternative to the quantile transform and the Hungarian construction. http://arxiv.org/abs/0711.0501 --------------------------------------------------------------- 6242. ON TIME DYNAMICS OF COAGULATION-FRAGMENTATION PROCESSES Boris L.Granovsky and Michael M. Erlihson We establish a characterization of coagulation-fragmentation processes, such that the induced birth and death processes depicting the total number of groups at time $t\ge 0$ are Markov and time homogeneous. Based on this, we provide a characterization of Gibbs coagulation-fragmentation models, which extends the one derived by Hendriks et al. As a by- product of our results, the class of solvable models is widened and two questions posed by N. Berestycki and Pitman are answered. http://arxiv.org/abs/0711.0503 --------------------------------------------------------------- 6243. FLUCTUATIONS FOR A CONSERVATIVE INTERFACE MODEL ON A WALL Lorenzo Zambotti We consider an effective interface model on a hard wall in (1+1) dimensions, with conservation of the area between the interface and the wall. We prove that the equilibrium fluctuations of the height variable converge in law to the solution of a SPDE with reflection and conservation of the space average. The proof is based on recent results obtained with L. Ambrosio and G. Savare on stability properties of Markov processes with log-concave invariant measures. http://arxiv.org/abs/0711.0583 --------------------------------------------------------------- 6244. ON THE STOCHASTIC BURGERS EQUATION WITH SOME APPLICATIONS TO TURBULENCE AND ASTROPHYSICS Andrew Neate and Aubrey Truman We summarise a selection of results on the inviscid limit of the stochastic Burgers equation emphasising geometric properties of the caustic, Maxwell set and Hamilton-Jacobi level surfaces and relating these results to a discussion of stochastic turbulence. We show that for small viscosities there exists a vortex filament structure near to the Maxwell set. We discuss how this vorticity is directly related to the adhesion model for the evolution of the early universe and include new explicit formulas for the distribution of mass within the shock. http://arxiv.org/abs/0711.0617 --------------------------------------------------------------- 6245. SKOROHOD-REFLECTION OF BROWNIAN PATHS AND BES^3 Balint Toth and Balint Veto Let B(t), X(t) and Y(t) be independent standard 1d Borwnian motions. Define X^+(t) and Y^-(t) as the trajectories of the processes X(t) and Y(t) pushed upwards and, respectively, downwards by B(t), according to Skorohod- reflection. In a recent paper, Jon Warren proves inter alia that Z(t):= X^+(t)-Y^- (t) is a three-dimensional Bessel-process. In this note, we present an alternative, elementary proof of this fact. http://arxiv.org/abs/0711.0631 --------------------------------------------------------------- 6246. HJB EQUATIONS FOR CERTAIN SINGULARLY CONTROLLED DIFFUSIONS Rami Atar and Amarjit Budhiraja and Ruth J. Williams Given a closed, bounded convex set $\mathcal{W}\subset{\mathbb {R}} ^d$ with nonempty interior, we consider a control problem in which the state process $W$ and the control process $U$ satisfy \[W_t= w_0+\int_0^t\vartheta(W_s) ds+\int_0^t\sigma(W_s) dZ_s+GU_t\in \mathcal{W},\qquad t\ge0,\] where $Z$ is a standard, multi-dimensional Brownian motion, $\vartheta,\sigma\in C^{0,1}(\mathcal{W})$, $G$ is a fixed matrix, and $w_0\in\mathcal{W} $. The process $U$ is locally of bounded variation and has increments in a given closed convex cone $\mathcal{U}\subset{\mathbb{R}}^p$. Given $g\in C(\mathcal{W})$, $\kappa\in{\mathbb{R}}^p$, and $\alpha>0$, consider the objective that is to minimize the cost \[J(w_0,U)\doteq\mathbb{E}\biggl[\int_0^{\infty}e^{-\alpha s}g(W_s) ds+\int_{[0,\infty)}e^{-\alpha s} d(\kappa\cdot U_s)\biggr]\] over the admissible controls $U$. Both $g$ and $\kappa\cdot u$ ($u\in\mathcal {U}$) may take positive and negative values. This paper studies the corresponding dynamic programming equation (DPE), a second-order degenerate elliptic partial differential equation of HJB-type with a state constraint boundary condition. Under the controllability condition $G\mathcal{U}={\mathbb{R}}^d$ and the finiteness of $\mathcal{H}(q)=\sup_{u\in\mathcal{U}_1}\{-Gu\cdot q- \kappa\cdot u\}$, $q\in {\mathbb{R}}^d$, where $\mathcal{U}_1=\{u\in\mathcal{U}:| Gu|=1\}$, we show that the cost, that involves an improper integral, is well defined. We establish the following: (i) the value function for the control problem satisfies the DPE (in the viscosity sense), and (ii) the condition $\inf_{q\in{\mathbb{R}}^d}\mathcal{H}(q)<0$ is necessary and sufficient for uniqueness of solutions to the DPE. The existence and uniqueness of solutions are shown to be connected to an intuitive ``no arbitrage'' condition. Our results apply to Brownian control problems that represent formal diffusion approximations to control problems associated with stochastic processing networks. http://arxiv.org/abs/0711.0641 --------------------------------------------------------------- 6247. SURVIVAL AND COMPLETE CONVERGENCE FOR A SPATIAL BRANCHING SYSTEM WITH LOCAL REGULATION Matthias Birkner and Andrej Depperschmidt We study a discrete time spatial branching system on $\mathbb{Z}^d $ with logistic-type local regulation at each deme depending on a weighted average of the population in neighboring demes. We show that the system survives for all time with positive probability if the competition term is small enough. For a restricted set of parameter values, we also obtain uniqueness of the nontrivial equilibrium and complete convergence, as well as long-term coexistence in a related two-type model. Along the way we classify the equilibria and their domain of attraction for the corresponding deterministic coupled map lattice on $\mathbb{Z}^d$. http://arxiv.org/abs/0711.0649 --------------------------------------------------------------- 6248. A CLASS OF SELF-SIMILAR STOCHASTIC PROCESSES WITH STATIONARY INCREMENTS TO MODEL ANOMALOUS DIFFUSION IN PHYSICS Antonio Mura and Francesco Mainardi In this paper we present a general mathematical construction that allows us to define a parametric class of $H$-sssi stochastic processes (self- similar with stationary increments), which have marginal probability density function that evolves in time according to a partial integro-differential equation of fractional type. This construction is based on the theory of finite measures on functional spaces. Since the variance evolves in time as a power function, these $H$-sssi processes naturally provide models for slow and fast anomalous diffusion. Such a class includes, as particular cases, fractional Brownian motion, grey Brownian motion and Brownian motion. http://arxiv.org/abs/0711.0665 --------------------------------------------------------------- 6249. DIFFERENTIAL EQUATIONS DRIVEN BY GAUSSIAN SIGNALS II Peter Friz and Nicolas Victoir Large classes of multi-dimensional Gaussian processes can be enhanced with stochastic Levy area(s). In a previous paper, we gave sufficient and essentially necessary conditions, only involving variational properties of the covariance. Following T. Lyons, the resulting lift to a "Gaussian rough path" gives a robust theory of (stochastic) differential equations driven by Gaussian signals with sample path regularity worse than Brownian motion. The purpose of this sequel paper is to establish convergence of Karhunen-Loeve approximations in rough path metrics. Particular care is necessary since martingale arguments are not enough to deal with third iterated integrals. An abstract support criterion for approximately continuous Wiener functionals then gives a description of the support of Gaussian rough paths as the closure of the (canonically lifted) Cameron-Martin space. http://arxiv.org/abs/0711.0668 --------------------------------------------------------------- 6250. MAXIMUM LIKELIHOOD ESTIMATORS AND RANDOM WALKS IN LONG MEMORY MODELS Karine Bertin and Soledad Torres and Ciprian Tudor (CES and SAMOS) We consider statistical models driven by Gaussian and non-Gaussian self-similar processes with long memory and we construct maximum likelihood estimators (MLE) for the drift parameter. Our approach is based on the approximation by random walks of the driving noise. We study the asymptotic behavior of the estimators and we give some numerical simulations to illustrate our results. http://arxiv.org/abs/0711.0513 --------------------------------------------------------------- 6251. CONFIRMATION OF MATHERON'S CONJECTURE ON THE COVARIOGRAM OF A PLANAR CONVEX BODY Gennadiy Averkov and Gabriele Bianchi The covariogram g_K of a convex body K in E^d is the function which associates to each x in E^d the volume of the intersection of K with K +x. In 1986 G. Matheron conjectured that for d=2 the covariogram g_K determines K within the class of all planar convex bodies, up to translations and reflections in a point. This problem is equivalent to some problems in stochastic geometry and probability as well as to a particular case of the phase retrieval problem in Fourier analysis. It is also relevant for the inverse problem of determining the atomic structure of a quasicrystal from its X-ray diffraction image. In this paper, using some results previously proved by the second named author, we confirm Matheron's conjecture completely. http://arxiv.org/abs/0711.0572 --------------------------------------------------------------- 6252. REFLECTED BSDE WITH QUADRATIC GROWTH AND UNBOUNDED TERMINAL VALUE J.-P. Lepeltier and M. Xu In this paper we prove the existence of a solution for reflected BSDE's\ whose coefficient is of quadratic growth in $z$ and of linear growth in $y$, with an unbounded terminal value. http://arxiv.org/abs/0711.0619 --------------------------------------------------------------- 6253. CORRECTION. PERFECT SIMULATION FOR A CLASS OF POSITIVE RECURRENT MARKOV CHAINS Stephen B. Connor and Wilfrid S. Kendall Correction to Annals of Applied Probability 17 (2007) 781--808 [doi:10.1214/105051607000000032]. http://arxiv.org/abs/0711.0804 --------------------------------------------------------------- 6254. WEIGHTED POWER VARIATIONS OF ITERATED BROWNIAN MOTION Ivan Nourdin (PMA) and Giovanni Peccati (LSTA) We characterize the asymptotic behaviour of the weighted power variation processes associated with iterated Brownian motion. We prove weak convergence results in the sense of finite dimensional distributions, and show that the laws of the limiting objects can always be expressed in terms of three independent Brownian motions X, Y and B, as well as of the local times of Y. In particular, our results involve "weighted'' versions of Kesten and Spitzer's Brownian motion in random scenery. Our findings extend the theory of stochastic integration developed theory initiated by Khoshnevisan and Lewis (1999), and should be compared with the recent results by Nourdin, Nualart and Tudor (2007) and Swanson (2007), concerning the weighted power variations of self- similar Gaussian processes. http://arxiv.org/abs/0711.0858 --------------------------------------------------------------- 6255. INFINITE VITERBI ALIGNMENTS IN THE TWO STATE HIDDEN MARKOV MODELS J. Lember and A. Koloydenko We show that, unlike in the general case, in the case of the two state HMM, the existence of infinite Viterbi alignments needs no special assumptions and can be proved considerably more easily. http://arxiv.org/abs/0711.0928 --------------------------------------------------------------- 6256. FIRST EXIT TIMES FOR L\'EVY-DRIVEN DIFFUSIONS WITH EXPONENTIALLY LIGHT JUMPS Peter Imkeller and Ilya Pavlyukevich and Torsten Wetzel We consider a dynamical system described by the differential equation dY_t=-U'(Y_t)dt with a unique stable point at the origin. We perturb the system by L\'evy noise of intensity \e, to obtain the stochastic differential equation dX^\e_t=-U'(X^\e_{t-})dt+\e dL_t. The process L is a symmetric L\'evy process whose jump measure \nu has exponentially light tails, \nu([u,\infty))\sim\exp(-u^\alpha), \alpha>0, u\to \infty. We study the first exit problem for the trajectories of the solutions of the stochastic differential equation from the interval (-1,1). In the small noise limit \e\to 0, the law of the first exit time \sigma_x, x\in(-1,1), is exponential with the mean value exhibiting an intriguing phase transition at the critical index \alpha=1, namely \log E \sigma\sim \e^{-\alpha} for 0<\alpha<1, whereas \log \E \sigma\sim \e^{-1}|\ln\e|^{1-\frac{1}{\alpha}} for \alpha>1. http://arxiv.org/abs/0711.0982 --------------------------------------------------------------- 6257. CONSTRUCTING PROCESSES WITH PRESCRIBED MIXING COEFFICIENTS Leonid (Aryeh) Kontorovich The rate at which dependencies between future and past observations decay in a random process may be quantified in terms of mixing coefficients. The latter in turn appear in strong laws of large numbers and concentration of measure results for dependent random variables. Questions regarding what rates are possible for various notions of mixing have been posed since the 1960's, and have important implications for some open problems in the theory of strong mixing conditions. This paper deals with $\eta$-mixing, a notion defined in [Kontorovich and Ramanan], which is closely related to $\phi$-mixing. We show that there exist measures on finite sequences with essentially arbitrary $\eta$-mixing coefficients, as well as processes with arbitrarily slow mixing rates. http://arxiv.org/abs/0711.0986 --------------------------------------------------------------- 6258. OBTAINING MEASURE CONCENTRATION FROM MARKOV CONTRACTION Leonid (Aryeh) Kontorovich Concentration bounds for non-product, non-Haar measures are fairly recent: the first such result was obtained for contracting Markov chains by Marton in 1996. Since then, several other such results have been proved; with few exceptions, these rely on coupling techniques. Though coupling is of unquestionable utility as a theoretical tool, it appears to have some limitations. Coupling has yet to be used to obtain bounds for more general Markov-type processes: hidden (or partially observed) Markov chains, Markov trees, etc. As an alternative to coupling, we apply the elementary Markov contraction lemma to obtain simple, useful, and apparently novel concentration results for the various Markov-type processes. Our technique consists of expressing probabilities as matrix products and applying Markov contraction to these expressions; thus it is fairly general and holds the potential to yield numerous results in this vein. http://arxiv.org/abs/0711.0987 --------------------------------------------------------------- 6259. A CONSERVATIVE EVOLUTION OF THE BROWNIAN EXCURSION Lorenzo Zambotti We consider the problem of conditioning the Brownian excursion to have a fixed time average over the interval [0,1] and we study an associated stochastic partial differential equation with reflection at 0 and with the constraint of conservation of the space average. The equation is driven by the derivative in space of a space-time white noise and contains a double Laplacian in the drift. Due to the lack of the maximum principle for the double Laplacian, the standard techniques based on the penalization method do not yield existence of a solution. http://arxiv.org/abs/0711.1068 --------------------------------------------------------------- 6260. MULTIVARIATE NORMAL APPROXIMATION WITH STEIN'S METHOD OF EXCHANGEABLE PAIRS UNDER A GENERAL LINEARITY CONDITION Gesine Reinert and Adrian R\"ollin We establish Stein's method of exchangeable pairs to assess distributional distances to potentially singular multivariate normal distributions, in terms of both smooth and non-smooth test functions. As examples we treat runs on the line, the joint count of edges, two-stars and triangles in Bernoulli random graphs, and complete $U$-statistics. Auxiliary random variables such as Hoeffding projections arise naturally in the construction of exchangeable pairs. http://arxiv.org/abs/0711.1082 --------------------------------------------------------------- 6261. AGING AND QUENCHED LOCALIZATION FOR ONE-DIMENSIONAL RANDOM WALKS IN RANDOM ENVIRONMENT IN THE SUB-BALLISTIC REGIME Nathana\"el Enriquez (MODAL'X and PMA) and Christophe Sabot (ICJ) and Olivier Zindy (WIAS) We consider transient one-dimensional random walks in random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of "valleys" of height $\log t$. In the quenched setting, we also sharply estimate the distribution of the walk at time $t$. http://arxiv.org/abs/0711.1095 --------------------------------------------------------------- 6262. APPROXIMATING PERPETUITIES Margarete Knape and Ralph Neininger We propose and analyze an algorithm to approximate distribution functions and densities of perpetuities. Our algorithm refines an earlier approach based on iterating discretized versions of the fixed point equation that defines the perpetuity. We significantly reduce the complexity of the earlier algorithm. Also one particular perpetuity arising in the analysis of the selection algorithm Quickselect is studied in more detail. Our approach works well for distribution functions. For densities we have weaker error bounds although computer experiments indicate that densities can also be approximated well. http://arxiv.org/abs/0711.1099 --------------------------------------------------------------- 6263. STRICT LOCAL MARTINGALES, BUBBLES, AND NO EARLY EXERCISE Soumik Pal and Philip Protter We show pathological behavior of asset price processes modeled by continuous strict local martingales under a risk-neutral measure. The inspiration comes from recent results on financial bubbles. We analyze, in particular, the effect of the strict nature of the local martingale on the usual formula for the price of a European call option, especially a strong anomaly when call prices decay monotonically with maturity. A complete and detailed analysis for the archetypical strict local martingale, the reciprocal of a three dimensional Bessel process, has been provided. Our main tool is based on a general h-transform technique (due to Delbaen and Schachermayer) to generate positive strict local martingales. This gives the basis for a statistical test to verify whether a suspected bubble is indeed one (or not). http://arxiv.org/abs/0711.1136 --------------------------------------------------------------- 6264. OPTIMAL INTERTEMPORAL RISK ALLOCATION APPLIED TO INSURANCE PRICING Kei Fukuda and Akihiko Inoue and Yumiharu Nakano We present a general approach to the pricing of products in finance and insurance in the multi-period setting. It is a combination of the utility indifference pricing and optimal intertemporal risk allocation. We give a characterization of the optimal intertemporal risk allocation by a first order condition. Applying this result to the exponential utility function, we obtain an essentially new type of premium calculation method for a popular type of multi-period insurance contract. This method is simple and can be easily implemented numerically. We see that the results of numerical calculations are well coincident with the risk loading level determined by traditional practices. The results also suggest a possible implied utility approach to insurance pricing. http://arxiv.org/abs/0711.1143 --------------------------------------------------------------- 6265. PROJECTIONS, ENTROPY AND SUMSETS Paul Balister and B\'ela Bollob\'as In this paper we have shall generalize Shearer's entropy inequality and its recent extensions by Madiman and Tetali, and shall apply projection inequalities to deduce extensions of some of the inequalities concerning sums of sets of integers proved recently by Gyarmati, Matolcsi and Ruzsa. We shall also discuss projection and entropy inequalities and their connections. http://arxiv.org/abs/0711.1151 --------------------------------------------------------------- 6266. CONTRIBUTIONS TO RANDOM ENERGY MODELS Nabin Kumar Jana In this thesis, we consider several Random Energy Models. This includes Derrida's Random Energy Model (REM) and Generalized Random Energy Model (GREM) and a nonhierarchical version (BK-GREM) by Bolthausen and Kistler. The limiting free energy in all these models along with Word GREM, a model proposed by us, turn out to be a cute consequence of large deviation principle (LDP). This LDP argument allows us to consider non-Gaussian driving distributions as well as external field. We could also consider random trees as the underlying tree structure in GREM. In all these models, as expected, limiting free energy is not 'universal' unlike the SK model. However it is 'rate specific'. Consideration of non-Gaussian driving distribution as well as different driving distributions for the different levels of the underlying trees in GREM leads to interesting phenomena. For example in REM, if the Hamiltonian is Binomial with parameter $N$ and $p$ then the existence of phase transition depends on the parameter $p$. More precisely, phase transition takes place only when $p>{1/2}$. For another example, consider a 2 level GREM with exponential driving distribution at the first level and Gaussian in the second with equal weights at both the levels. Then even if the limiting ratio for the second level particles, $p_2$ is 0.00001 (very small), the system reduces to a Gaussian REM. On the other hand, if we consider a 2 level GREM with Gaussian driving distribution at the first level and exponential in the second, the system will never reduce to a Gaussian REM. In either case, the system will never reduce to that of an exponential REM. etc. http://arxiv.org/abs/0711.1249 --------------------------------------------------------------- 6267. THE CRITICAL CONTACT PROCESS IN A RANDOMLY EVOLVING ENVIRONMENT DIES OUT Jeffrey E. Steif and Marcus Warfheimer Bezuidenhout and Grimmett proved that the critical contact process dies out. Here, we generalize the result to the so called contact process in a random evolving environment (CPREE), introduced by Erik Broman. This process is a generalization of the contact process where the recovery rate can vary between two values. The rate which it chooses is determined by a background process, which evolves independently at different sites. As for the contact process, we can similarly define a critical value in terms of survival for this process. In this paper we prove that this definition is independent of how we start the background process, that finite and infinite survival (meaning nontriviality of the upper invariant measure) are equivalent and finally that the process dies out at criticality. http://arxiv.org/abs/0711.1258 --------------------------------------------------------------- 6268. HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACK-MERTON-SCHOLES? Walter Schachermayer and Josef Teichmann We compare the option pricing formulas of Louis Bachelier and Black-Merton-Scholes and observe -- theoretically as well as for Bachelier's original data -- that the prices coincide very well. We illustrate Louis Bachelier's efforts to obtain applicable formulas for option pricing in pre-computer time. Furthermore we explain -- by simple methods from chaos expansion -- why Bachelier's model yields good short-time approximations of prices and volatilities. http://arxiv.org/abs/0711.1272 --------------------------------------------------------------- 6269. LOCAL PROBABILITIES FOR RANDOM WALKS CONDITIONED TO STAY POSITIVE Vladimir Vatutin and Vitali Wachtel Let S_0=0,{S_n, n>0} be a random walk generated by a sequence of i.i.d. random variables X_1,X_2,... and let \tau^{-} be the first descending ladder epoch. Assuming that the distribution of X_1 belongs to the domain of attraction of an \alpha-stable law we study the asymptotic behavior of the local probabilities P(\tau ^{-}=n) and the conditional local probabilities P(S_n\in [x,x+y)|\tau^{-}>n) for fixed y and x=x(n)\in (0,\infty). http://arxiv.org/abs/0711.1302 --------------------------------------------------------------- 6270. FRACTIONAL MARTINGALES AND CHARACTERIZATION OF THE FRACTIONAL BROWNIAN MOTION Yaozhong Hu and David Nualart and Jian Song In this paper we introduce the notion of $\alpha$-martingale as the fractional derivative of order $\alpha$ of a continuous local martingale, where $ \alpha\in (-\frac 12, \frac 12)$, and we show that it has a nonzero finite variation of order $\frac 2{1+2\alpha}$, under some integrability assumptions on the quadratic variation of the local martingale. As an application we establish an extension of L\'evy's characterization theorem for the fractional Brownian motion. http://arxiv.org/abs/0711.1313 --------------------------------------------------------------- 6271. FROM RANDOM MATRICES TO RANDOM ANALYTIC FUNCTIONS Manjunath Krishnapur We consider two families of random matrix-valued analytic functions: (1) G_1-zG_2 and (2) G_0 + zG_1 +z^2G_2+ ..., where G_i are n x n independent random matrices with independent standard complex Gaussian entries. The set of z where these matrix-valued analytic functions become singular, are shown to be determinantal point processes on the sphere and the hyperbolic plane, respectively. The kernels of these determinantal processes are reproducing kernels of certain natural Hilbert spaces of analytic functions on the corresponding surfaces. This gives a unified framework in which to view a result of Peres and Virag (n=1 in the second setting) and a well known theorem of Ginibre on Gaussian random matrices (which may be viewed as an analogue of our results in the whole plane). http://arxiv.org/abs/0711.1378 --------------------------------------------------------------- 6272. ON WEIGHTED APPROXIMATIONS IN $D[0, 1]$ WITH APPLICATIONS TO SELF-NORMALIZED PARTIAL SUM PROCESSES Mikl\'os Cs\"org\H{o} and Barbara Szyszkowicz and Qiying Wang Let $X, X_1, X_2,...$ be a sequence of non-degenerate i.i.d. random variables with mean zero. The best possible weighted approximations are investigated in $D[0, 1]$ for the partial sum processes $\{S_{[nt]}, 0\le t\le 1\}$, where $S_n=\sum_{j=1}^nX_j$, under the assumption that $X$ belongs to the domain of attraction of the normal law. The conclusions then are used to establish similar results for the sequence of self-normalized partial sum processes $\{S_{[nt]}/V_n, 0\le t\le 1\}$, where $V_n^2=\sum_{j=1}^nX_j^2$. $L_p$ approximations of self-normalized partial sum processes are also discussed. http://arxiv.org/abs/0711.1384 --------------------------------------------------------------- 6273. ASYMPTOTICS OF STUDENTIZED U-TYPE PROCESSES FOR CHANGEPOINT PROBLEMS Mikl\'os Cs\"org\H{o} and Barbara Szyszkowicz and Qiying Wang This paper investigates weighted approximations for studentized $U$-statistics type processes, both with symmetric and antisymmetric kernels, only under the assumption that the distribution of the projection variate is in the domain of attraction of the normal law. The classical second moment condition $E|h(X_1,X_2)|^2 < \infty$ is also relaxed in both cases. The results can be used for testing the null assumption of having a random sample versus the alternative that there is a change in distribution in the sequence. http://arxiv.org/abs/0711.1385 --------------------------------------------------------------- 6274. WEAK CONVERGENCE OF ERROR PROCESSES IN DISCRETIZATIONS OF STOCHASTIC INTEGRALS AND BESOV SPACES Stefan Geiss and Anni Toivola We consider the weak convergence of the rescaled error processes for Riemann discretizations of certain stochastic integrals and relate the integrability of their weak limit to the fractional smoothness of the stochastic integral. http://arxiv.org/abs/0711.1439 --------------------------------------------------------------- 6275. ON WEAK TAIL DOMINATION OF RANDOM VECTORS Rafa{\l} Lata{\l}a Motivated by a question of Krzysztof Oleszkiewicz we study a notion of weak tail domination of random vectors. We show that if the dominating random variable is sufficiently regular weak tail domination implies strong tail domination. In particular positive answer to Oleszkiewicz question would follow from the so-called Bernoulli conjecture. http://arxiv.org/abs/0711.1477 --------------------------------------------------------------- 6276. TWO BESSEL BRIDGES CONDITIONED NEVER TO COLLIDE, DOUBLE DIRICHLET SERIES, AND JACOBI THETA FUNCTION Makoto Katori and Minami Izumi and Naoki Kobayashi It is known that the moments of the maximum value of a one- dimensional conditional Brownian motion, the three-dimensional Bessel bridge with duration 1 started from the origin, are expressed using the Riemann zeta function. We consider a system of two Bessel bridges, in which noncolliding condition is imposed. We show that the moments of the maximum value is then expressed using the double Dirichlet series, or using the integrals of products of the Jacobi theta functions and its derivatives. Since the present system will be provided as a diffusion scaling limit of a version of vicious walker model, the ensemble of 2-watermelons with a wall, the dominant terms in long-time asymptotics of moments of height of 2-watermelons are completely determined, for which only the first moment, i.e. the average height, was recently studied by Fulmek by a method of enumerative combinatorics. http://arxiv.org/abs/0711.1710 --------------------------------------------------------------- 6277. SOME SHORT PROOFS FOR CONNECTEDNESS OF BOUNDARIES Adam Timar We generalize theorems of Kesten and Deuschel-Pisztora about the connectedness of the exterior boundary of a connected subset of $\Z^d $, where "connectedness" and "boundary" are understood with respect to various graphs on the vertices of $\Z^d$. We provide simple and elementary proofs of their results. It turns out that the proper way of viewing these questions is graph theory, instead of topology. http://arxiv.org/abs/0711.1713 --------------------------------------------------------------- 6278. RENYI INFORMATION FOR ERGODIC DIFFUSION PROCESSES Alessandro De Gregorio and Stefano Iacus In this paper we derive explicit formulas of the R\'enyi information, Shannon entropy and Song measure for the invariant density of one dimensional ergodic diffusion processes. In particular, the diffusion models considered include the hyperbolic, the generalized inverse Gaussian, the Pearson, the exponential familiy and a new class of skew-$t$ diffusions. http://arxiv.org/abs/0711.1789 --------------------------------------------------------------- 6279. ON THE ASYMPTOTIC BEHAVIOUR OF INCREASING POSITIVE SELF- SIMILAR MARKOV PROCESSES Maria Emilia Caballero and Victor Rivero We are interested by the rate of growth of increasing positive self-similar Markov processes (ipssMp) such that the subordinator associated to it via Lamperti's transformation has infinite mean. We prove that the logarithm of an ipssMp normalized by the logarithm of the time converges weakly, as the time tends to infinity, if and only if the Laplace exponent of the underlying subordinator is regularly varying at zero. Moreover, we prove that the regular variation at zero of the Laplace exponent is essentially nasc for the existence of a function that normalizes the logarithm of an ipssMp. We obtain a law of iterated logarithm for the liminf of the logarithm of an ipssMp and an integral test to study the upper envelope of it. Furthermore, results concerning the rate of growth of the random clock appearing in Lamperti's transformation are obtained. http://arxiv.org/abs/0711.1834 --------------------------------------------------------------- 6280. A MEASURABLE-GROUP-THEORETIC SOLUTION TO VON NEUMANN'S PROBLEM Damien Gaboriau (UMPA-ENSL) and Russell Lyons We give a positive answer, in the measurable-group-theory context, to von Neumann's problem of knowing whether a non-amenable countable discrete group contains a non-cyclic free subgroup. We also get an embedding result of the free-group von Neumann factor into restricted wreath product factors. http://arxiv.org/abs/0711.1643 --------------------------------------------------------------- 6281. CUTSETS IN INFINITE GRAPHS Adam Timar We answer three questions posed in a paper by Babson and Benjamini. They introduced a parameter $C_G$ for Cayley graphs $G$ that has significant application to percolation. For a minimal cutset of $G$ and a partition of this cutset into two classes, take the minimal distance between the two classes. The supremum of this number over all minimal cutsets and all partitions is $C_G$. We show that if it is finite for some Cayley graph of the group then it is finite for any (finitely generated) Cayley graph. Having an exponential bound for the number of minimal cutsets of size $n$ separating $o$ from infinity also turns out to be independent of the Cayley graph chosen. We show a 1- ended example (the lamplighter group), where $C_G$ is infinite. Finally, we give a new proof for a question of de la Harpe, proving that the number of $n $-element cutsets separating $o$ from infinity is finite unless $G$ is a finite extension of $Z$. http://arxiv.org/abs/0711.1711 --------------------------------------------------------------- 6282. NUMBER VARIANCE OF RANDOM ZEROS ON COMPLEX MANIFOLDS, II: SMOOTH STATISTICS Bernard Shiffman and Steve Zelditch We consider the zero sets $Z_N$ of systems of $m$ random polynomials of degree $N$ in $m$ complex variables, and we give asymptotic formulas for the random variables given by summing a smooth test function over $Z_N$. Our asymptotic formulas show that the variances for these smooth statistics have the growth $N^{m-2}$. We also prove analogues for the integrals of smooth test forms over the subvarieties defined by $k4$, and appropriate versions of $\SLE_{\hat\kappa}$, $\hat\kappa=16/\kappa$. http://arxiv.org/abs/0711.1884 --------------------------------------------------------------- 6284. A CLASS OF INFINITE DIMENSIONAL DIFFUSION PROCESSES WITH CONNECTION TO POPULATION GENETICS Shui Feng and Feng-Yu Wang Starting from a sequence of independent Wright-Fisher diffusion processes on $[0,1]$, we construct a class of reversible infinite dimensional diffusion processes on $\DD_\infty:= \{{\bf x}\in [0,1]^\N: \sum_{i\ge 1} x_i=1 \}$ with GEM distribution as the reversible measure. Log-Sobolev inequalities are established for these diffusions, which lead to the exponential convergence to the corresponding reversible measures in the entropy. Extensions are made to a class of measure-valued processes over an abstract space $S$. This provides a reasonable alternative to the Fleming-Viot process which does not satisfy the log-Sobolev inequality when $S$ is infinite as observed by W. Stannat \cite{S}. http://arxiv.org/abs/0711.1887 --------------------------------------------------------------- 6285. GROWTH OF THE NUMBER OF SPANNING TREES OF THE ERD\"OS-R\'ENYI GIANT COMPONENT Russell Lyons and Ron Peled and Oded Schramm The number of spanning trees in the giant component of the random graph $\G(n, c/n)$ ($c>1$) grows like $\exp\big\{m\big(f(c)+o(1)\big)\big\} $ as $n\to\infty$, where $m$ is the number of vertices in the giant component. The function $f$ is not known explicitly, but we show that it is strictly increasing and infinitely differentiable. Moreover, we give an explicit lower bound on $f'(c)$. A key lemma is the following. Let $\PGW(\lambda)$ denote a Galton-Watson tree having Poisson offspring distribution with parameter $\lambda$. Suppose that $\lambda^*>\lambda>1$. We show that $\PGW (\lambda^*)$ conditioned to survive forever stochastically dominates $\PGW(\lambda)$ conditioned to survive forever. http://arxiv.org/abs/0711.1893 --------------------------------------------------------------- 6286. A LOCAL TIME CORRESPONDENCE FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS Mohammud Foondun and Davar Khoshnevisan and Eulalia Nualart It is frequently the case that a white-noise-driven parabolic and/or hyperbolic stochastic partial differential equation (SPDE) can have random-field solutions only in spatial dimension one. Here we show that in many cases, where the ``spatial operator'' is the L^2-generator of a L \'evy process X, a linear SPDE has a random-field solution if and only if the symmetrization of X possesses local times. This result gives a probabilistic reason for the lack of existence of random-field solutions in dimensions strictly bigger than one. In addition, we prove that the solution to the SPDE is [H\"older] continuous in its spatial variable if and only if the said local time is [H\"older] continuous in its spatial variable. We also produce examples where the random-field solution exists, but is almost surely unbounded in every open subset of space-time. Our results are based on first establishing a quasi-isometry between the linear L^2-space of the weak solutions of a family of linear SPDEs, on one hand, and the Dirichlet space generated by the symmetrization of X, on the other hand. We study mainly linear equations in order to present the local-time correspondence at a modest technical level. However, some of our work has consequences for nonlinear SPDEs as well. We demonstrate this assertion by studying a family of parabolic SPDEs that have additive nonlinearities. For those equations we prove that if the linearized problem has a random-field solution, then so does the nonlinear SPDE. Moreover, the solution to the linearized equation is [H\"older] continuous if and only if the solution to the nonlinear equation is. And the solutions are bounded and unbounded together as well. Finally, we prove that in the cases that the solutions are unbounded, they almost surely blow up at exactly the same points. http://arxiv.org/abs/0711.1913 --------------------------------------------------------------- 6287. SPLITTING FOR RARE EVENT SIMULATION: A LARGE DEVIATION APPROACH TO DESIGN AND ANALYSIS Thomas Dean and Paul Dupuis Particle splitting methods are considered for the estimation of rare events. The probability of interest is that a Markov process first enters a set $B$ before another set $A$, and it is assumed that this probability satisfies a large deviation scaling. A notion of subsolution is defined for the related calculus of variations problem, and two main results are proved under mild conditions. The first is that the number of particles generated by the algorithm grows subexponentially if and only if a certain scalar multiple of the importance function is a subsolution. The second is that, under the same condition, the variance of the algorithm is characterized (asymptotically) in terms of the subsolution. The design of asymptotically optimal schemes is discussed, and numerical examples are presented. http://arxiv.org/abs/0711.2037 --------------------------------------------------------------- 6288. MARTINGALE DIMENSIONS FOR FRACTALS Masanori Hino We prove that the martingale dimensions for canonical diffusion processes on a class of self-similar sets including nested fractals are always one. This provides an affirmative answer to the conjecture of S. Kusuoka [Publ. Res. Inst. Math. Sci. 25 (1989) 659--680]. http://arxiv.org/abs/0711.2135 --------------------------------------------------------------- 6289. A SINGULAR CONTROL MODEL WITH APPLICATION TO THE GOODWILL PROBLEM Andrew J. F. Jack and Timothy C. Johnson and Mihail Zervos We consider a stochastic system whose uncontrolled state dynamics are modelled by a general one-dimensional It\^{o} diffusion. The control effort that can be applied to this system takes the form that is associated with the so-called monotone follower problem of singular stochastic control. The control problem that we address aims at maximising a performance criterion that rewards high values of the utility derived from the system's controlled state but penalises any expenditure of control effort. This problem has been motivated by applications such as the so-called goodwill problem in which the system's state is used to represent the image that a product has in a market, while control expenditure is associated with raising the product's image, e.g., through advertising. We obtain the solution to the optimisation problem that we consider in a closed analytic form under rather general assumptions. Also, our analysis establishes a number of results that are concerned with analytic as well as probabilistic expressions for the first derivative of the solution to a second order linear non-homogeneous ordinary differential equation. These results have independent interest and can potentially be of use to the solution of other one-dimensional stochastic control problems. http://arxiv.org/abs/0711.2143 --------------------------------------------------------------- 6290. REFLECTING ORNSTEIN-UHLENBECK PROCESSES ON PINNED PATH SPACES Masanori Hino and Hiroto Uchida Consider a set of continuous maps from the interval $[0,1]$ to a domain in ${\mathbb R}^d$. Although the topological boundary of this set in the path space is not smooth in general, by using the theory of functions of bounded variation (BV functions) on the Wiener space and the theory of Dirichlet forms, we can discuss the existence of the surface measure and the Skorokhod representation of the reflecting Ornstein-Uhlenbeck process associated with the canonical Dirichlet form on this set. http://arxiv.org/abs/0711.2144 --------------------------------------------------------------- 6291. THE KEY RENEWAL THEOREM FOR A TRANSIENT MARKOV CHAIN Dmitry Korshunov We consider a time-homogeneous Markov chain $X_n$, $n\ge0$, valued in ${\bf R}$. Suppose that this chain is transient, that is, $X_n$ generates a $\sigma$-finite renewal measure. We prove the key renewal theorem under condition that this chain has asymptotically homogeneous at infinity jumps and asymptotically positive drift. http://arxiv.org/abs/0711.2169 --------------------------------------------------------------- 6292. EXACT FINITE APPROXIMATIONS OF AVERAGE-COST COUNTABLE MARKOV DECISION PROCESSES Arie Leizarowitz and Adam Shwartz For a countable-state Markov decision process we introduce an embedding which produces a finite-state Markov decision process. The finite-state embedded process has the same optimal cost, and moreover, it has the same dynamics as the original process when restricting to the approximating set. The embedded process can be used as an approximation which, being finite, is more convenient for computation and implementation. http://arxiv.org/abs/0711.2185 --------------------------------------------------------------- 6293. EFFICIENT ROUTING IN HEAVY TRAFFIC UNDER PARTIAL SAMPLING OF SERVICE TIMES Rami Atar and Adam Shwartz We consider a queue with renewal arrivals and n exponential servers in the Halfin-Whitt heavy traffic regime, where n and the arrival rate increase without bound, so that a critical loading condition holds. Server k serves at rate $\mu_k $, and the empirical distribution of the $\mu_k $ is assumed to converge weakly. We show that very little information on the service rates is required for a routing mechanism to perform well. More precisely, we construct a routing mechanism that has access to a single sample from the service time distribution of each of $n$ to the power of $1/2 + \epsilon $ randomly selected servers, but not to the actual values of the service rates, the performance of which is asymptotically as good as the best among mechanisms that have the complete information on $ \mu_k $. http://arxiv.org/abs/0711.2188 --------------------------------------------------------------- 6294. UNIQUENESS OF A CONSTRAINED VARIATIONAL PROBLEM AND LARGE DEVIATIONS OF BUFFER SIZE Adam Shwartz and Alan Weiss We show global uniqueness of the solution to a class of constrained variational problems, using scaling properties. This is used to establish the essential uniqueness of solutions of a large deviations problem in multiple dimensions. The result is motivated by models of buffers, and in particular the probability of, and typical path to overflow in the limit of small buffers, which we analyze. http://arxiv.org/abs/0711.2191 --------------------------------------------------------------- 6295. THE AIZENMAN-SIMS-STARR SCHEME FOR THE SK MODEL WITH MULTIDIMENSIONAL SPINS Anton Bovier and Anton Klimovsky The non-hierarchical correlation structure of the Sherrington- Kirkpatrick (SK) model with multidimensional (e.g. Heisenberg) spins is studied at the level of the logarithmic asymptotic of the corresponding sum of the correlated exponentials -- the thermodynamic pressure. For this purpose an abstract quenched large deviations principle (LDP) of Gaertner-Ellis type is obtained under an assumption of measure concentration. With the aid of this principle the framework of the Aizenman-Sims-Starr comparison scheme ($\text{AS} ^2$ scheme) is extended to the case of the SK model with multidimensional spins. This extension, based the quenched LDP, shows how the Hadamard matrix products arise rigorously in the context of the Parisi formula. This allows one to relate the pressure of the non-hierarchical SK model with the pressure of the hierarchical GREM by a saddle-point variational formula of the Parisi type including a negative remainder term. http://arxiv.org/abs/0711.2286 --------------------------------------------------------------- 6296. SPIRAL MODEL: A CELLULAR AUTOMATON WITH A DISCONTINUOUS GLASS TRANSITION Cristina Toninelli and Giulio Biroli We introduce a new class of two-dimensional cellular automata with a bootstrap percolation-like dynamics. Each site can be either empty or occupied by a single particle and the dynamics follows a deterministic updating rule at discrete times which allows only emptying sites. We prove that the threshold density $\rho_c$ for convergence to a completely empty configuration is non trivial, $0<\rho_c<1$, contrary to standard bootstrap percolation. Furthermore we prove that in the subcritical regime, $\rho<\rho_c$, emptying always occurs exponentially fast and that $\rho_c$ coincides with the critical density for two-dimensional oriented site percolation on $\bZ^2$. This is known to occur also for some cellular automata with oriented rules for which the transition is continuous in the value of the asymptotic density and the crossover length determining finite size effects diverges as a power law when the critical density is approached from below. Instead for our model we prove that the transition is {\it discontinuous} and at the same time the crossover length diverges {\it faster than any power law}. The proofs of the discontinuity and the lower bound on the crossover length use a conjecture on the critical behaviour for oriented percolation. The latter is supported by several numerical simulations and by analytical (though non rigorous) works through renormalization techniques. Finally, we will discuss why, due to the peculiar {\it mixed critical/first order character} of this transition, the model is particularly relevant to study glassy and jamming transitions. Indeed, we will show that it leads to a dynamical glass transition for a Kinetically Constrained Spin Model. Most of the results that we present are the rigorous proofs of physical arguments developed in a joint work with D.S.Fisher. http://arxiv.org/abs/0709.0378 --------------------------------------------------------------- 6297. ON THE INFORMATION RATES OF THE PLENOPTIC FUNCTION Arthur Cunha and Minh Do and and Martin Vetterli The {\it plenoptic function} (Adelson and Bergen, 91) describes the visual information available to an observer at any point in space and time. Samples of the plenoptic function (POF) are seen in video and in general visual content, and represent large amounts of information. In this paper we propose a stochastic model to study the compression limits of the plenoptic function. In the proposed framework, we isolate the two fundamental sources of information in the POF: the one representing the camera motion and the other representing the information complexity of the ``reality'' being acquired and transmitted. The sources of information are combined, generating a stochastic process that we study in detail. We first propose a model for ensembles of realities that do not change over time. The proposed model is simple in that it enables us to derive precise coding bounds in the information-theoretic sense that are sharp in a number of cases of practical interest. For this simple case of static realities and camera motion, our results indicate that coding practice is in accordance with optimal coding from an information-theoretic standpoint. The model is further extended to account for visual realities that change over time. We derive bounds on the lossless and lossy information rates for this dynamic reality model, stating conditions under which the bounds are tight. Examples with synthetic sources suggest that in the presence of scene dynamics, simple hybrid coding using motion/displacement estimation with DPCM performs considerably suboptimally relative to the true rate-distortion bound. http://arxiv.org/abs/0711.2104 --------------------------------------------------------------- 6298. MEAN-FIELD BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS. A LIMIT APPROACH Rainer Buckdahn and Juan Li and Shige Peng Mathematical mean-field approaches play an important role in different fields of Physics and Chemistry, but have found in recent works also their application in Economics, Finance and Game Theory. The objective of our paper is to study a special mean-field problem in a purely stochastic approach. We consider a stochastic differential equation that describes the dynamics of a particle $X^{(N)}$ influenced by the dynamics of $N$ other particles, which are supposed to be independent identically distributed and of the same law as $X^ {(N)}$. This equation (of rank $N$) is then associated with a backward stochastic differential equation (BSDE). After proving the existence and the uniqueness of a solution $(X^{(N)},Y^{(N)},Z^{(N)})$ for this couple of equations we investigate its limit behavior. With an approach which uses the tightness of the laws of the above sequence of triplets in a suitable space, and combines it with BSDE methods and the Law of Large Numbers, it is shown that $(X^{(N)},Y^{(N)},Z^{(N)})$ converges in $L^2$ to the unique solution of a limit equation, formed by a Mean-Field forward and a Mean-Field backward equation. http://arxiv.org/abs/0711.2162 --------------------------------------------------------------- 6299. URN-RELATED RANDOM WALK WITH DRIFT $\RHO X^{\ALPHA} / T^{\BETA}$ Mikhail Menshikov and Stanislav Volkov We study a one-dimensional random walk whose expected drift depends both on time and the position of a particle. We establish a non-trivial phase transition for the recurrence vs. transience of the walk, and show some interesting applications to Friedman's urn, as well as showing the connection with Lamperti's walk with asymptotically zero drift. http://arxiv.org/abs/0711.2373 --------------------------------------------------------------- 6300. POISSON APPROXIMATION FOR SEARCH OF RARE WORDS IN DNA SEQUENCES Nicolas Vergne (1) and Miguel Abadi (2) ((1) Laboratoire Statistique et G\'enome France, (2) Universidade de Campinas Brazil) Using recent results on the occurrence times of a string of symbols in a stochastic process with mixing properties, we present a new method for the search of rare words in biological sequences generally modelled by a Markov chain. We obtain a bound on the error between the distribution of the number of occurrences of a word in a sequence (under a Markov model) and its Poisson approximation. A global bound is already given by a Chen-Stein method. Our approach, the psi-mixing method, gives local bounds. Since we only need the error in the tails of distribution, the global uniform bound of Chen- Stein is too large and it is a better way to consider local bounds. We search for two thresholds on the number of occurrences from which we can regard the studied word as an over-represented or an under-represented one. A biological role is suggested for these over- or under-represented words. Our method gives such thresholds for a panel of words much broader than the Chen-Stein method. Comparing the methods, we observe a better accuracy for the psi- mixing method for the bound of the tails of distribution. We also present the software PANOW (available at http://stat.genopole.cnrs.fr/software/panowdir/) dedicated to the computation of the error term and the thresholds for a studied word. http://arxiv.org/abs/0711.2382 --------------------------------------------------------------- 6301. A SUFFICIENT CONDITION TO DETERMINE ATOMS OF A SIGMA ALGEBRA VIA ITS GENERATOR Jinshan Zhang To constitute atoms of a sigma algebra is not a easy job due to the large number of its elements. Thus, determining them via the generator seems a feasible and simple way since most sigma algebras are generated by their smaller proper subsets. Precisely, Under some conditions each atom of a sigma algebra equals the intersection of the elememts containing any point of the atom in the generator. In this paper, a very weak sufficient condition for determining atoms by the generator will be presented. Besides, such a condition, though not a necessary one, will be shown to be almost the weakest one, say, almost can not be improved. http://arxiv.org/abs/0711.2400 --------------------------------------------------------------- 6302. ORNSTEIN-UHLENBECK PROCESSES ON LIE GROUPS Fabrice Baudoin and Martin Hairer and Josef Teichmann We consider Ornstein-Uhlenbeck processes (OU-processes) related to hypoelliptic diffusion on finite-dimensional Lie groups: let $ \mathcal{L} $ be a hypoelliptic, left-invariant ``sum of the squares''-operator on a Lie group $ G $ with associated Markov process $ X $, then we construct OU-type processes by adding horizontal gradient drifts of functions $ U $. In the natural case $ U(x) = - \log p(1,x) $, where $ p(1,x) $ is the density of the law of the Markov process $ X $ starting at the identity $ e $ at time $ t =1 $ with respect to the right-invariant Haar measure on $G$, we show the Poincar\'e inequality by applying the Driver-Melcher inequality for ``sum of the squares'' operators on Lie groups. The Markov process associated to $ - \log p(1,x) $ is called the OU-process related to the given hypoelliptic diffusion on $ G $. We prove the global strong existence of this OU-process on $ G $. The Poincar\'e inequality for a large class of potentials $U$ is then shown by perturbation methods and used to obtain a hypoelliptic equivalent of the standard result on cooling schedules for simulated annealing. The relation between local results on $ \mathcal{L} $ and global results for the constructed OU-process is widely used in this study. http://arxiv.org/abs/0711.2419 --------------------------------------------------------------- 6303. A TWO-DIMENSIONAL RUIN PROBLEM ON THE POSITIVE QUADRANT Florin Avram and Zbigniew Palmowski and Martijn Pistorius In this paper we study the joint ruin problem for two insurance companies that divide between them both claims and premia in some specified proportions (modeling two branches of the same insurance company or an insurance and re-insurance company). Modeling the risk processes of the insurance companies by Cram\'{e}r-Lundberg processes we obtain the Laplace transform in space of the probability that either of the insurance companies is ruined in finite time. Subsequently, for exponentially distributed claims, we derive an explicit analytical expression for this joint ruin probability by explicitly inverting this Laplace transform. We also provide a characterization of the Laplace transform of the joint ruin time. http://arxiv.org/abs/0711.2465 --------------------------------------------------------------- 6304. COPULAS: COMPATIBILITY AND FR\'ECHET CLASSES Fabrizio Durante and Erich Peter Klement and Jos\'e Juan Quesada- Molina We determine under which conditions three bivariate copulas are compatible, viz. they are the bivariate marginals of the same trivariate copula, and, then, construct the class of these copulas. In particular, the upper and lower bounds for this class of trivariate copulas are determined. http://arxiv.org/abs/0711.2409 --------------------------------------------------------------- 6305. A SINGULAR STOCHASTIC DIFFERENTIAL EQUATION DRIVEN BY FRACTIONAL BROWNIAN MOTION Yaozhong Hu and David Nualart and Xiaoming Song In this paper we study a singular stochastic differential equation driven by an additive fractional Brownian motion with Hurst parameter $H>\frac 12$. Under some assumptions on the drift, we show that there is a unique solution, which has moments of all orders. We also apply the techniques of Malliavin calculus to prove that the solution has an absolutely continuous law at any time $t>0$. http://arxiv.org/abs/0711.2507 --------------------------------------------------------------- 6306. A SHORT NOTE ON SMALL DEVIATIONS OF SEQUENCES OF I.I.D. RANDOM VARIABLES WITH EXPONENTIALLY DECREASING WEIGHTS Frank Aurzada We obtain some new results concerning the small deviation problem for $S=\sum_n q^n X_n$ and $M=\sup_n q^n X_n$, where $01/3. http://arxiv.org/abs/0711.2633 --------------------------------------------------------------- 6308. A NOTE ON RANDOM WALKS IN A HYPERCUBE Stanislav Volkov and Timothy Wong We study a simple random walk on an n-dimensional hypercube. For any starting position we find the probability of hitting vertex a before hitting vertex b, whenever a and b share the same edge. This generalizes the model in Doyle, P., and Snell, J., "Random Walks and Electric Networks", Mathematical Association of America, 1984 (see Exercise 1.3.7 there). http://arxiv.org/abs/0711.2675 --------------------------------------------------------------- 6309. ON THE RANK OF RANDOM SPARSE MATRICES Kevin P. Costello and Van Vu We investigate the rank of random (symmetric) sparse matrices. Our main finding is that with high probability, any dependency that occurs in such a matrix is formed by a set of few rows that contains an overwhelming number of zeros. This allows us to obtain an exact estimate for the co-rank. http://arxiv.org/abs/0711.2696 --------------------------------------------------------------- 6310. THE LARGEST SAMPLE EIGENVALUE DISTRIBUTION IN THE RANK 1 QUATERNIONIC SPIKED MODEL OF WISHART ENSEMBLE Dong Wang We solve the largest sample eigenvalue distribution problem in the rank 1 spiked model of the quaternionic Wishart ensemble, which is the first case of a statistical generalization of the Laguerre symplectic ensemble (LSE) on the soft edge. We observe a phase change phenomenon similar to that in the complex case, and prove that the new distribution at the phase change point is the GOE Tracy-Widom distribution. http://arxiv.org/abs/0711.2722 --------------------------------------------------------------- 6311. FREE MARTINGALE POLYNOMIALS FOR STATIONARY JACOBI PROCESSES Nizar Demni (PMA) We generalize a previous result concerning free martingale polynomials for the stationary free Jacobi process of parameters $\lambda \in ]0.1], \theta = 1/2$. Hopelessly, apart from the case $\lambda = 1$, the polynomials we derive are no longer orthogonal with respect to the spectral measure. As a matter of fact, we use the multiplicative renormalization to write down the corresponding orthogonality measure. http://arxiv.org/abs/0711.2734 --------------------------------------------------------------- 6312. PRICING EQUITY DEFAULT SWAPS UNDER AN APPROXIMATION TO THE CGMY L\'{E}% VY MODEL Soeren Asmussen and Dilip Madan and Martijn Pistorius The Wiener-Hopf factorization is obtained in closed form for a phase type approximation to the CGMY L\'{e}vy process. This allows, for the approximation, exact computation of first passage times to barrier levels via Laplace transform inversion. Calibration of the CGMY model to market option prices defines the risk neutral process for which we infer the first passage times of stock prices to 30% of the price level at contract initiation. These distributions are then used in pricing 50% recovery rate equity default swap (EDS) contracts and the resulting prices are compared with the prices of credit default swaps (CDS). An illustrative analysis is presented for these contracts on Ford and GM. http://arxiv.org/abs/0711.2807 --------------------------------------------------------------- 6313. G-BROWNIAN MOTION AND DYNAMIC RISK MEASURE UNDER VOLATILITY UNCERTAINTY Shige Peng We introduce a new notion of G-normal distributions. This will bring us to a new framework of stochastic calculus of Ito's type (Ito's integral, Ito's formula, Ito's equation) through the corresponding G-Brownian motion. We will also present analytical calculations and some new statistical methods with application to risk analysis in finance under volatility uncertainty. Our basic point of view is: sublinear expectation theory is very like its special situation of linear expectation in the classical probability theory. Under a sublinear expectation space we still can introduce the notion of distributions, of random variables, as well as the notions of joint distributions, marginal distributions, etc. A particularly interesting phenomenon in sublinear situations is that a random variable Y is independent to X does not automatically implies that X is independent to Y. Two important theorems have been proved: The law of large number and the central limit theorem. http://arxiv.org/abs/0711.2834 --------------------------------------------------------------- 6314. A FAMILY OF MARTINGALES GENERATED BY A PROCESS WITH INDEPENDENT INCREMENTS Josep Llu\'is Sol\'e and Frederic Utzet An explicit procedure to construct a family of martingales generated by a process with independent increments is presented. The main tools are the polynomials that give the relationship between the moments and cumulants, and a set of martingales related to the jumps of the process called Teugels martingales http://arxiv.org/abs/0711.2879 --------------------------------------------------------------- 6315. STOCHASTIC MECHANICS AS A GAUGE THEORY Claudio Albanese We introduce a classical diffusion process which provides a full description of non-relativistic quantum mechanics and has the form of a Z_4 gauge theory. We first define a stochastic process on a discretization of physical space of the form (aZ)^3, where a is an elementary length scale. We then lift this process to the principal bundle (aZ)^3 x Z_4. Non-relativistic quantum mechanics is recovered in the limit as a tends to 0, as we show in the case of a scalar particle in an electromagnetic field. Many-body interactions can easily be accommodated. In the case of tight binding Hamiltonians no limit needs to be taken, the equivalence is straightforward and sheds new light on the dynamics of quantum phases. http://arxiv.org/abs/0711.2978 --------------------------------------------------------------- 6316. STOCHASTIC INTEGRALS AND ABELIAN PROCESSES Claudio Albanese We study triangulation schemes for the joint kernel of a diffusion process with uniformly continuous coefficients and an adapted, non-resonant Abelian process. The prototypical example of Abelian process to which our methods apply is given by stochastic integrals with uniformly continuous coeffcients. The range of applicability includes also a broader class of processes of practical relevance, such as the sup process and certain discrete time summations we discuss. We discretize the space coordinate in uniform steps and assume that time is either continuous or finely discretized as in a fully explicit Euler method and the Courant condition is satisfied. We show that the Fourier transform of the joint kernel of a diffusion and a stochastic integral converges in a uniform graph norm associated to the Markov generator. Convergence also implies smoothness properties for the Fourier transform of the joint kernel. Stochastic integrals are straightforward to define for finite triangulations and the convergence result gives a new and entirely constructive way of defining stochastic integrals in the continuum. The method relies on a reinterpretation and extension of the classic theorems by Feynman-Kac, Girsanov, Ito and Cameron-Martin, which are also re-obtained. We make use of a path-wise analysis without relying on a probabilistic interpretation. The Fourier representation is needed to regularize the hypo-elliptic character of the joint process of a diffusion and an adapted stochastic integral. The argument extends as long as the Fourier analysis framework can be generalized. This condition leads to the notion of non-resonant Abelian process. http://arxiv.org/abs/0711.2980 --------------------------------------------------------------- 6317. INVERSE SAMPLING FOR NONASYMPTOTIC SEQUENTIAL ESTIMATION OF BOUNDED VARIABLE MEANS Xinjia Chen In this paper, we consider the nonasymptotic sequential estimation of means of random variables bounded in between zero and one. We have rigorously demonstrated that, in order to guarantee prescribed relative precision and confidence level, it suffices to continue sampling until the sample sum is no less than a certain bound and then take the average of samples as an estimate for the mean of the bounded random variable. We have developed an explicit formula and a bisection search method for the determination of such bound of sample sum, without any knowledge of the bounded variable. Moreover, we have derived bounds for the distribution of sample size. In the special case of Bernoulli random variables, we have established analytical and numerical methods to further reduce the bound of sample sum and thus improve the efficiency of sampling. http://arxiv.org/abs/0711.2801 --------------------------------------------------------------- 6318. ENERGY DISCRIMINANT ANALYSIS, QUANTUM LOGIC, AND FUZZY SETS Grigorii Melnichenko It is shown that the quantum logic of linear subspaces can be used for recognition of random signals by a Bayesian energy discriminant classifier. The energy distribution on linear subspaces is described by a correlation matrix of probability distribution. We show that the correlation matrix corresponds to von Neumann density matrix in the quantum theory. We offered the interpretation of quantum logic as a fuzzy logic of fuzzy sets. The use of quantum logic for recognition is based on the fact that the probability distribution of each class lies approximately on a lower-dimensional subspace of feature space. It is offered interpretation of discriminant functions as membership functions of fuzzy sets. Also we offer the quality functional for optimal choose of discriminant functions for recognition from some class of discriminant functions. http://arxiv.org/abs/0711.1437 --------------------------------------------------------------- 6319. AUTOMORPHISM GROUPS OF FINITE P-GROUPS: STRUCTURE AND APPLICATIONS Geir T. Helleloid This thesis has three goals related to the automorphism groups of finite $p$-groups. The primary goal is to provide a complete proof of a theorem showing that, in some asymptotic sense, the automorphism group of almost every finite $p$-group is itself a $p$-group. We originally proved this theorem in a paper with Martin; the presentation of the proof here contains omitted proof details and revised exposition. We also give a survey of the extant results on automorphism groups of finite $p$-groups, focusing on the order of the automorphism groups and on known examples. Finally, we explore a connection between automorphisms of finite $p$-groups and Markov chains. Specifically, we define a family of Markov chains on an elementary abelian $p$-group and bound the convergence rate of some of those chains. http://arxiv.org/abs/0711.2816 --------------------------------------------------------------- 6320. POSITIVE ASSOCIATION IN THE FRACTIONAL FUZZY POTTS MODEL Jeff Kahn and Nicholas Weininger A fractional fuzzy Potts measure is a probability distribution on spin configurations of a finite graph $G$ obtained in two steps: first a subgraph of $G$ is chosen according to a random cluster measure $\phi_{p,q}$, and then a spin ($\pm1$) is chosen independently for each component of the subgraph and assigned to all vertices of that component. We show that whenever $q \geq1$, such a measure is positively associated, meaning that any two increasing events are positively correlated. This generalizes earlier results of H\"{a}ggstr\"{o}m [Ann. Appl. Probab. 9 (1999) 1149--1159] and H\"{a}ggstr\"{o}m and Schramm [Stochastic Process. Appl. 96 (2001) 213--242]. http://arxiv.org/abs/0711.3136 --------------------------------------------------------------- 6321. BOUNDARY PROXIMITY OF SLE Oded Schramm and Wang Zhou This paper examines how close the chordal $\SLE_\kappa$ curve gets to the real line asymptotically far away from its starting point. In particular, when $\kappa\in(0,4)$, it is shown that if $\beta>\beta_\kappa:=1/(8/ \kappa-2)$, then the intersection of the $\SLE_\kappa$ curve with the graph of the function $y=x/(\log x)^{\beta}$, $x>e$, is a.s. bounded, while it is a.s. unbounded if $\beta=\beta_\kappa$. The critical $\SLE_4$ curve a.s. intersects the graph of $y=x^{-(\log\log x)^\alpha}$, $x>e^e$, in an unbounded set if $\alpha \le 1$, but not if $\alpha>1$. Under a very mild regularity assumption on the function $y(x)$, we give a necessary and sufficient integrability condition for the intersection of the $\SLE_\kappa$ path with the graph of $y$ to be unbounded. We also prove that the Hausdorff dimension of the intersection set of the $\SLE_{\kappa}$ curve and real axis is $2-8/\kappa$ when $4<\kappa<8$. http://arxiv.org/abs/0711.3350 --------------------------------------------------------------- 6322. LINEAR LOWER BOUNDS FOR $\DELTA_C(P)$ FOR A CLASS OF 2D SELF- DESTRUCTIVE PERCOLATION MODELS J. van den Berg and B.N.B. de Lima The self-destructive percolation model is defined as follows: Consider percolation with parameter $p > p_c$. Remove the infinite occupied cluster. Finally, give each vertex (or, for bond percolation, each edge) that at this stage is vacant, an extra chance $\delta$ to become occupied. Let $ \delta_c(p)$ be the minimal value of $\delta$, needed to obtain an infinite occupied cluster in the final configuration. This model was introduced some years ago by van den Berg and Brouwer. They showed that, for the site model on the square lattice (and a few other 2D lattices satisfying a special technical condition) that $\delta_c(p)\geq\frac{(p-p_c)}{p}$. In particular, $\delta_c(p)$ is at least linear in $p-p_c$. Although the arguments used by van den Berg and Brouwer look quite rigid, we show that they can be suitably modified to obtain similar linear lower bounds for $\delta_c(p)$ (with $p$ near $p_c$) for a much larger class of 2D lattices, including bond percolation on the square and triangular lattices, and site percolation on the star lattice (or matching lattice) of the square lattice. http://arxiv.org/abs/0711.3563 --------------------------------------------------------------- 6323. STOCHASTIC DOMINATION FOR A HIDDEN MARKOV CHAIN WITH APPLICATIONS TO THE CONTACT PROCESS IN A RANDOMLY EVOLVING ENVIRONMENT Erik I. Broman The ordinary contact process is used to model the spread of a disease in a population. In this model, each infected individual waits an exponentially distributed time with parameter 1 before becoming healthy. In this paper, we introduce and study the contact process in a randomly evolving environment. Here we associate to every individual an independent two-state, $\{0,1 \},$ background process. Given $\delta_0<\delta_1,$ if the background process is in state $0,$ the individual (if infected) becomes healthy at rate $ \delta_0,$ while if the background process is in state $1,$ it becomes healthy at rate $\delta_1.$ By stochastically comparing the contact process in a randomly evolving environment to the ordinary contact process, we will investigate matters of extinction and that of weak and strong survival. A key step in our analysis is to obtain stochastic domination results between certain point processes. We do this by starting out in a discrete setting and then taking continuous time limits. http://arxiv.org/abs/0711.3597 --------------------------------------------------------------- 6324. RECONSTRUCTION FOR COLORINGS ON TREES Nayantara Bhatnagar and Juan Vera and and Eric Vigoda Consider $k$-colorings of the complete tree of depth $\ell$ and branching factor $\Delta$. If we fix the coloring of the leaves, for what range of $k$ is the root uniformly distributed over all $k$ colors (in the limit $\ell\to\infty$)? This corresponds to the threshold for uniqueness of the infinite-volume Gibbs measure. It is straightforward to show the existence of colorings of the leaves which ``freeze'' the entire tree when $k\le \Delta+1$. For $k\geq\Delta+2$, Jonasson proved the root is ``unbiased'' for any fixed coloring of the leaves and thus the Gibbs measure is unique. What happens for a {\em typical} coloring of the leaves? When the leaves have a non- vanishing influence on the root in expectation, over random colorings of the leaves, reconstruction is said to hold. Non-reconstruction is equivalent to extremality of the Gibbs measure. When $k<\Delta/\ln{\Delta}$, it is straightforward to show that reconstruction is possible (and hence the measure is not extremal). We prove that for $C>2$ and $k =C\Delta/\ln{\Delta}$, non- reconstruction holds, i.e., the Gibbs measure is extremal. We prove a strong form of extremality: with high probability over the colorings of the leaves the influence at the root decays exponentially fast with the depth of the tree. These are the first results coming close to the threshold for extremality for colorings. Extremality on trees and random graphs has received considerable attention recently since it may have connections to the efficiency of local algorithms. http://arxiv.org/abs/0711.3664 --------------------------------------------------------------- 6325. STRONG INVARIANCE PRINCIPLES FOR DEPENDENT RANDOM VARIABLES Wei Biao Wu We establish strong invariance principles for sums of stationary and ergodic processes with nearly optimal bounds. Applications to linear and some nonlinear processes are discussed. Strong laws of large numbers and laws of the iterated logarithm are also obtained under easily verifiable conditions. http://arxiv.org/abs/0711.3674 --------------------------------------------------------------- 6326. LIMIT LAWS FOR BIASED RANDOM WALKS ON A GALTON-WATSON TREE WITH LEAVES Alexander Fribergh (ICJ) and Nina Gantert We consider an outwardly $\beta$-biased random walk $X_n$ on a Galton-Watson tree with leaves in the sub-ballistic regime. We prove that $X_n/n^ {\gamma}$ convergences in law and we characterize the limit law. The exponent $ \gamma\in (0,1)$ is explicit and is a decreasing function of $\beta$. Key tools for the proof are classical decomposition results for Galton-Watson trees, a new variant of regeneration times and the careful analysis of the time the walker spends in leaves. http://arxiv.org/abs/0711.3686 --------------------------------------------------------------- 6327. THE POSTERIOR METRIC AND THE GOODNESS OF GIBBSIANNESS FOR TRANSFORMS OF GIBBS MEASURES C. Kuelske and A. A. Opoku We present a general method to derive continuity estimates for conditional probabilities of general (possibly continuous) spin models sub jected to local transformations. Such systems arise in the study of a stochastic time- evolution of Gibbs measures or as noisy observations. We exhibit the minimal necessary structure for such double-layer systems. Assuming no a priori metric on the local state spaces, we define the posterior metric on the local image space. We show that it allows in a natural way to divide the local part of the continuity estimates from the spatial part (which is treated by Dobrushin uniqueness here). We show in the concrete example of the time evolution of rotators on the q-1 dimensional sphere how this method can be used to obtain estimates in terms of the familiar Euclidean metric. http://arxiv.org/abs/0711.3764 --------------------------------------------------------------- 6328. THE LNDELOF HYPOTHESIS FOR ALMOST ALL HURWITZ'S ZETA-FUNCTIONS HOLDS TRUE Masumi Nakajima By Probability theory, that is, by a kind of quasi-law of the iterated logarithm, we prove the title claim. http://arxiv.org/abs/0711.3784 --------------------------------------------------------------- 6329. FREE BROWNIAN MOTION AND EVOLUTION TOWARDS BOXPLUS-INFINITE DIVISIBILITY FOR K-TUPLES Serban T. Belinschi and Alexandru Nica Let D be the space of non-commutative distributions of k-tuples of selfadjoints in a C*-probability space (for a fixed k). We introduce a semigroup of transformations B_t of D, such that every distribution in D evolves under the B_t towards infinite divisibility with respect to free additive convolution. The very good properties of B_t come from some special connections that we put into evidence between free additive convolution and the operation of Boolean convolution. On the other hand we put into evidence a relation between the transformations B_t and free Brownian motion. More precisely, we introduce a transformation Phi of D which converts the free Brownian motion started at an arbitrary distribution m in D into the process B_t (Phi(m)), t>0. http://arxiv.org/abs/0711.3787 --------------------------------------------------------------- 6330. A PDE FOR THE MULTI-TIME JOINT PROBABILITY OF THE AIRY PROCESS Dong Wang This paper gives a PDE for multi-time joint probability of the Airy process, which generalizes Adler and van Moerbeke's result on the 2-time case. As an intermediate step, the PDE for the multi-time joint probability of the Dyson Brownian motion is also given. http://arxiv.org/abs/0711.3797 --------------------------------------------------------------- 6331. HYPERFINITE GRAPH LIMITS Oded Schramm G\'abor Elek introduced the notion of a hyperfinite graph family: a collection of graphs is hypefinite if for every $\epsilon>0$ there is some finite $k$ such that each graph $G$ in the collection can be broken into connected components of size at most $k$ by removing a set of edges of size at most $\epsilon|V(G)|$. We presently extend this notion to a certain compactification of finite bounded-degree graphs, and show that if a sequence of finite graphs converges to a hyperfinite limit, then the sequence itself is hyperfinite. http://arxiv.org/abs/0711.3808 --------------------------------------------------------------- 6332. THE STRUCTURE OF THE ALLELIC PARTITION OF THE TOTAL POPULATION FOR GALTON-WATSON PROCESSES WITH NEUTRAL MUTATIONS Jean Bertoin (DMA and Pma) We consider a (sub) critical Galton-Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We specify the law of this allelic partition in terms of the distribution of the number of clone- children and the number of mutant-children of a typical individual. The approach combines an extension of Harris representation of Galton-Watson processes and a version of the ballot theorem. Some limit theorems related to the distribution of the allelic partition are also given. http://arxiv.org/abs/0711.3852 --------------------------------------------------------------- 6333. FORWARD ESTIMATION FOR ERGODIC TIME SERIES Gusztav Morvai and Benjamin Weiss The forward estimation problem for stationary and ergodic time series $\{X_n\}_{n=0}^{\infty}$ taking values from a finite alphabet ${\cal X}$ is to estimate the probability that $X_{n+1}=x$ based on the observations $X_i$, $0\le i\le n$ without prior knowledge of the distribution of the process $\{X_n\}$. We present a simple procedure $g_n$ which is evaluated on the data segment $(X_0,...,X_n)$ and for which, ${\rm error}(n) = |g_{n}(x)-P (X_{n+1}=x |X_0,...,X_n)|\to 0$ almost surely for a subclass of all stationary and ergodic time series, while for the full class the Cesaro average of the error tends to zero almost surely and moreover, the error tends to zero in probability. http://arxiv.org/abs/0711.3856 --------------------------------------------------------------- 6334. THE INTERACTION BETWEEN MULTI-OVERLAPS IN THE HIGH TEMPERATURE PHASE OF THE SHERRINGTON-KIRKPATRICK SPIN GLASS Nicholas Crawford We explore the joint behavior of a finite number of multi- overlaps in the high temperature phase of the SK model. Extending work by M. Tala- grand, we show that, when these objects are scaled to have non-trivial limiting distributions, the joint behavior is described by a Gaussian process with an explicit covariance structure. http://arxiv.org/abs/0711.3873 --------------------------------------------------------------- 6335. MODERATE DEVIATIONS FOR STATIONARY SEQUENCES OF BOUNDED RANDOM VARIABLES J\'er\^ome Dedecker (LSTA) and Florence Merlev\`ede (PMA) and Magda Peligrad, Sergey Utev In this paper we derive the moderate deviation principle for stationary sequences of bounded random variables under martingale-type conditions. Applications to functions of $\phi$-mixing sequences, contracting Markov chains, expanding maps of the interval, and symmetric random walks on the circle are given. http://arxiv.org/abs/0711.3924 --------------------------------------------------------------- 6336. PARKING ON A RANDOM TREE H. Dehling and S. R. Fleurke and C. Kuelske Consider an infinite tree with random degrees, i.i.d. over the sites, with a prescribed probability distribution with generating function G(s). We consider the following variation of Renyi's parking problem, alternatively called blocking RSA: at every vertex of the tree a particle (or car) arrives with rate one. The particle sticks to the vertex whenever the vertex and all of its nearest neighbors are not occupied yet. We provide an explicit expression for the so-called parking constant in terms of the generating function. http://arxiv.org/abs/0711.4061 --------------------------------------------------------------- 6337. HAUSDORFF DIMENSION OF THE SLE CURVE INTERSECTED WITH THE REAL LINE Tom Alberts and Scott Sheffield We establish an upper bound on the asymptotic probability of an SLE (kappa) curve hitting two small intervals on the real line as the interval width goes to zero, for the range 4 < kappa < 8. As a consequence we are able to prove that the SLE curve intersected with the real line has Hausdorff dimension 2-8/kappa, almost surely. http://arxiv.org/abs/0711.4070 --------------------------------------------------------------- 6338. GIBBSIANNESS VERSUS NON-GIBBSIANNESS OF TIME-EVOLVED PLANAR ROTOR MODELS A.C.D. van Enter and W.M.Ruszel We study the Gibbsian character of time-evolved planar rotor systems on Z^d, d at least 2, in the transient regime, evolving with stochastic dynamics and starting with an initial Gibbs measure. We model the system by interacting Brownian diffusions, moving on circles. We prove that for small times and arbitrary initial Gibbs measures \nu, or for long times and both high- or infinite-temperature measure and dynamics, the evolved measure \nu^t stays Gibbsian. Furthermore we show that for a low-temperature initial measure \nu, evolving under infinite-temperature dynamics thee is a time interval (t_0, t_1) such that \nu^t fails to be Gibbsian in d=2. http://arxiv.org/abs/0711.3621 --------------------------------------------------------------- 6339. QUENCHED CLT FOR RANDOM TORAL AUTOMORPHISM Arvind Ayyer and Carlangelo Liverani and Mikko Stenlund We establish a quenched Central Limit Theorem (CLT) for a smooth observable of random sequences of iterated linear hyperbolic maps on the torus. To this end we also obtain an annealed CLT for the same system. We show that, almost surely, the variance of the quenched system is the same as for the annealed system. Our technique is the study of the transfer operator on an anisotropic Banach space specifically tailored to use the cone condition satisfied by the maps. http://arxiv.org/abs/0711.3818 --------------------------------------------------------------- 6340. MODELING SNOW CRYSTAL GROWTH III: THREE-DIMENSIONAL SNOWFAKES Janko Gravner and David Griffeath We introduce a three-dimensional, computationally feasible, mesoscopic model for snow crystal growth, based on diffusion of vapor, anisotropic attachment, and a semi-liquid boundary layer. Several case studies are presented that faithfully emulate a wide variety of physical snowflakes. http://arxiv.org/abs/0711.4020 --------------------------------------------------------------- 6341. MEAN DENSITY OF INHOMOGENEOUS BOOLEAN MODELS WITH LOWER DIMENSIONAL TYPICAL GRAIN Elena Villa The mean density of a random closed set $\Theta$ in $R^d$ with Hausdorff dimension $n$ is the Radon-Nikodym derivative of the expected measure $E[H^n(\Theta\cap\cdot)]$ induced by $\Theta$ with respect to the usual $d$-dimensional Lebesgue measure. We consider here inhomogeneous Boolean models with lower dimensional typical grain. Under general regularity assumptions on the typical grain, related to the existence of its Minkowski content, and on the intensity measure of the underlying Poisson point process, we provide an explicit formula for the mean density. Particular cases and examples are also discussed. Moreover, an estimator of the mean density naturally arises in terms of the empirical capacity functional, which turns to be closely related to the well known random variable density estimation by histograms in the extreme case $n=0$. http://arxiv.org/abs/0711.4202 --------------------------------------------------------------- 6342. AN INEQUALITY FOR CORRELATED MEASURABLE FUNCTIONS Fabio Zucca A classical inequality, which is known for families of monotone functions, is generalized to a larger class of families of measurable functions. Moreover we characterize all the families of functions for which the equality holds. We apply this result to a problem arising from probability theory. http://arxiv.org/abs/0711.4127 --------------------------------------------------------------- 6343. ON THREE DIFFERENT NOTIONS OF MONOTONE SUBSEQUENCES Miklos Bona We review how the monotone pattern compares to other patterns in terms of enumerative results on pattern avoiding permutations. We consider three natural definitions of pattern avoidance, give an overview of classic and recent formulas, and provide some new results related to limiting distributions. http://arxiv.org/abs/0711.4325 --------------------------------------------------------------- 6344. RECURRENT EXTENSIONS OF SELF-SIMILAR MARKOV PROCESSES AND CRAM \'ER'S CONDITION II V\'ictor Rivero We prove that a positive self-similar Markov process $(X,\mathbb {P})$ that hits 0 in a finite time admits a self-similar recurrent extension that leaves 0 continuously if and only if the underlying L\'{e}vy process satisfies Cram\'{e}r's condition. http://arxiv.org/abs/0711.4442 --------------------------------------------------------------- 6345. A QUENCHED LIMIT THEOREM FOR THE LOCAL TIME OF RANDOM WALKS ON \Z^2 J\"urgen G\"artner and Rongfeng Sun Let $X$ and $Y$ be two independent random walks on $\Z^2$ with zero mean and finite variances, and let $L_t(X,Y)$ be the local time of $X-Y$ at the origin at time $t$. We show that almost surely with respect to $Y$, $L_t (X,Y)/\log t$ conditioned on $Y$ converges in distribution to an exponential random variable with the same mean as the distributional limit of $L_t(X,Y)/\log t$ without conditioning. This question arises naturally from the study of the parabolic Anderson model with a single moving catalyst, which is closely related to a pinning model. http://arxiv.org/abs/0711.4488 --------------------------------------------------------------- 6346. LOWER LIMITS FOR DISTRIBUTIONS OF RANDOMLY STOPPED SUMS Denis Denisov and Serguei Foss and Dmitry Korshunov We study lower limits for the ratio $\frac{\bar{F^{*\tau}}(x)} {\bar F(x)}$ of tail distributions where $ F^{*\tau}$ is a distribution of a sum of a random size $\tau$ of i.i.d. random variables having a common distribution $F $, and a random variable $\tau$ does not depend on summands. http://arxiv.org/abs/0711.4491 --------------------------------------------------------------- 6347. INTEGRATED HARNACK INEQUALITIES ON LIE GROUPS Bruce K. Driver and Maria Gordina We prove an integrated Harnack inequality for heat kernels on uni- modular Lie groups. A key feature of these inequalities is that they only involve a constant depending on a lower bound for the Ricci curvature tensor. In particular, they are independent of dimension and hence are applicable in infinite--dimensional settings. http://arxiv.org/abs/0711.4392 --------------------------------------------------------------- 6348. HIERARCHICAL PINNING MODELS, QUADRATIC MAPS AND QUENCHED DISORDER Giambattista Giacomin and Hubert Lacoin and Fabio Lucio Toninelli We consider a hierarchical model of polymer pinning in presence of quenched disorder, introduced by B. Derrida, V. Hakim and J. Vannimenius in 1992, which can be re-interpreted as an infinite dimensional dynamical system with random initial condition (the disorder). It is defined through a recurrence relation for the law of a random variable {R_n}_{n=1,2,...}, which in absence of disorder (i.e., when the initial condition is degenerate) reduces to a particular case of the well-known Logistic Map. The large-n limit of the sequence of random variables 2^{-n} log R_n, a non-random quantity which is naturally interpreted as a free energy, plays a central role in our analysis. The model depends on a parameter alpha>0, related to the geometry of the hierarchical lattice, and has a phase transition in the sense that the free energy is positive if the expectation of R_0 is larger than a certain threshold value, and it is zero otherwise. It was conjectured by Derrida et al. (1992) that disorder is relevant (respectively, irrelevant or marginally relevant) if 1/21/2 we find the correct scaling form (for weak disorder) of the critical point shift. http://arxiv.org/abs/0711.4649 --------------------------------------------------------------- 6349. LOCAL INDEPENDENCE OF FRACTIONAL BROWNIAN MOTION Ilkka Norros and Eero Saksman Let S(t,t') be the sigma-algebra generated by the differences X(s)- X(s) with s,s' in the interval(t,t'), where (X_t) is the fractional Brownian motion process with Hurst index H between 0 and 1. We prove that for any two distinct t and t' the sigma-algebras S(t-a,t+a) and S(t'-a,t'+a) are asymptotically independent as a tends to 0. We show this in the strong sense that Shannon's mutual information between these two sigma-algebras tends to zero as a tends to 0. Some generalizations and quantitative estimates are provided also. http://arxiv.org/abs/0711.4809 --------------------------------------------------------------- 6350. H\"OLDER-DIFFERENTIABILITY OF GIBBS DISTRIBUTION FUNCTIONS Marc Kesseb\"ohmer and Bernd O. Stratmann In this paper we give non-trivial applications of the thermodynamic formalism to the theory of distribution functions of Gibbs measures (devil's staircases) supported on limit sets of finitely generated conformal iterated function systems in $\R$. For a large class of these Gibbs states we determine the Hausdorff dimension of the set of points at which the distribution function of these measures is not $\alpha$-H\"older-differentiable. The obtained results give significant extensions of recent work by Darst, Dekking, Falconer, Li, Morris, and Xiao. In particular, our results clearly show that the results of these authors have their natural home within thermodynamic formalism. http://arxiv.org/abs/0711.4698 --------------------------------------------------------------- 6351. A RANDOM WALK ON Z WITH DRIFT DRIVEN BY ITS OCCUPATION TIME AT ZERO Iddo Ben-Ari and Mathieu Merle and Alexander Roitershtein We consider a nearest neighbor random walk on the one-dimensional integer lattice with drift towards the origin determined by an asymptotically vanishing function of the number of visits to zero. We show the existence of distinct regimes according to the rate of decay of the drift. In particular, when the rate is sufficiently slow, the position of the random walk, properly normalized, converges to a symmetric exponential law. In this regime, in contrast to the classical case, the range of the walk scales differently from its position. http://arxiv.org/abs/0711.4871 --------------------------------------------------------------- 6352. LARGE DEVIATIONS FOR RANDOM WALK IN A SPACE-TIME PRODUCT ENVIRONMENT Atilla Yilmaz We consider random walk $(X_n)_{n\geq0}$ on $\mathbb{Z}^d$ in a space-time product environment $\omega\in\Omega$. We take the point of view of the particle and focus on the environment Markov chain $(T_{n,X_n}\omega)_ {n\geq0}$ where $T$ denotes the shift on $\Omega$. Conditioned on the particle having asymptotic speed equal to any given $\xi$, we show that the environment Markov chain converges to a stationary process $\mu_\xi$ under the annealed measure. When $d\geq3$ and $\xi$ is sufficiently close to the typical speed, we prove that annealed and quenched large deviations are equivalent and when conditioned on the particle having asymptotic speed $\xi$, the environment Markov chain converges to $\mu_\xi$ under the quenched measure as well. In this case, we show that $\mu_\xi$ is a stationary Markov process whose kernel is obtained from the original kernel by a Doob transform. http://arxiv.org/abs/0711.4872 --------------------------------------------------------------- 6353. NEAR-CRITICAL PERCOLATION IN TWO DIMENSIONS Pierre Nolin (LM-Orsay and DMA) We give a self-contained and detailed presentation of Kesten's results that allow to relate critical and near-critical percolation on the triangular lattice. They constitute an important step in the derivation of the exponents describing the near-critical behavior of this model. For future use and reference, we also show how these results can be obtained in more general situations, and we state some new consequences. http://arxiv.org/abs/0711.4948 --------------------------------------------------------------- 6354. COUPLING TIMES WITH AMBIGUITIES FOR PARTICLE SYSTEMS AND APPLICATIONS TO CONTEXT-DEPENDENT DNA SUBSTITUTION MODELS Jean B\'erard and Didier Piau We define a notion of coupling time with ambiguities for interacting particle systems, and show how this can be used to prove ergodicity and to bound the convergence time to equilibrium and the decay of correlations at equilibrium. A motivation is to provide simple conditions which ensure that perturbed particle systems share some properties of the underlying unperturbed system. We apply these results to context-dependent substitution models recently introduced by molecular biologists as descriptions of DNA evolution processes. These models take into account the influence of the neighboring bases on the substitution probabilities at a site of the DNA sequence, as opposed to most usual substitution models which assume that sites evolve independently of each other. http://arxiv.org/abs/0712.0072 --------------------------------------------------------------- 6355. ON ESTIMATING THE MEMORY FOR FINITARILY MARKOVIAN PROCESSES Gusztav Morvai and Benjamin Weiss Finitarily Markovian processes are those processes $\{X_n\}_{n=-\infty}^{\infty}$ for which there is a finite $K$ ($K = K(\{X_n\}_{n=-\infty}^0$) such that the conditional distribution of $X_1$ given the entire past is equal to the conditional distribution of $X_1$ given only $\{X_n\}_{n=1-K}^0$. The least such value of $K$ is called the memory length. We give a rather complete analysis of the problems of universally estimating the least such value of $K$, both in the backward sense that we have just described and in the forward sense, where one observes successive values of $\{X_n\}$ for $n \geq 0$ and asks for the least value $K$ such that the conditional distribution of $X_{n+1}$ given $\{X_i\}_{i=n-K+1}^n$ is the same as the conditional distribution of $X_{n+1}$ given $\{X_i\}_{i=- \infty}^n$. We allow for finite or countably infinite alphabet size. http://arxiv.org/abs/0712.0105 --------------------------------------------------------------- 6356. GENERATING FUNCTIONS OF CAUCHY-STIELTJES TYPE FOR ORTHOGONAL POLYNOMIALS Marek Bozejko and Nizar Demni We characterize by the use of free probability the family of measures for which the mulitiplicative renormalization method applies with $h(x) = (1-x)^_{-1}$. This provides a representation formula for their Voiculescu Transforms. http://arxiv.org/abs/0712.0156 --------------------------------------------------------------- 6357. THE LIMITS OF NESTED SUBCLASSES OF SEVERAL CLASSES OF INFINITELY DIVISIBLE DISTRIBUTIONS ARE IDENTICAL WITH THE CLOSURE OF THE CLASS OF STABLE DISTRIBUTIONS Makoto Maejima and Ken-iti Sato It is shown that the limits of the nested subclasses of five classes of infinitely divisible distributions on $R^d$, which are the Jurek class, the Goldie-Steutel-Bondesson class, the class of selfdecomposable distributions, the Thorin class and the class of generalized type $G$ distributions, are identical with the closure of the class of stable distributions. More general results are also given. http://arxiv.org/abs/0712.0206 --------------------------------------------------------------- 6358. A BIRTHDAY PARADOX FOR MARKOV CHAINS, WITH AN OPTIMAL BOUND FOR COLLISION IN THE POLLARD RHO ALGORITHM FOR DISCRETE LOGARITHM Jeong Han Kim and Ravi Montenegro and Yuval Peres and and Prasad Tetali We show a Birthday Paradox for self-intersections of Markov chains with uniform stationary distribution. As an application, we analyze Pollard's Rho algorithm for finding the discrete logarithm in a cyclic group G and find that, if the partition in the algorithm is given by a random oracle, then with high probability a collision occurs in order |G|^0.5 steps. This is the first proof of the correct order bound which does not assume that every step of the algorithm produces an i.i.d. sample from G. http://arxiv.org/abs/0712.0220 --------------------------------------------------------------- 6359. LYAPUNOV CONDITIONS FOR LOGARITHMIC SOBOLEV AND SUPER POINCAR \'E INEQUALITY Patrick Cattiaux (CMAP and MODAL'X) and Arnaud Guillin (LATP) and Feng-Yu Wang, Liming Wu We show how to use Lyapunov functions to obtain functional inequalities which are stronger than Poincar\'e inequality (for instance logarithmic Sobolev or $F$-Sobolev). The case of Poincar\'e and weak Poincar\'e inequalities was studied in Bakry and al. This approach allows us to recover and extend in an unified way some known criteria in the euclidean case (Bakry-Emery, Wang, Kusuoka-Stroock ...). http://arxiv.org/abs/0712.0235 --------------------------------------------------------------- 6360. RANDOM GRAPHS WITH FORBIDDEN VERTEX DEGREES Geoffrey Grimmett and Svante Janson We study the random graph G_{n,\lambda/n} conditioned on the event that all vertex degrees lie in some given subset S of the non-negative integers. Subject to a certain hypothesis on S, the empirical distribution of the vertex degrees is asymptotically Poisson with some parameter \mux given as the root of a certain `characteristic equation' of S that maximises a certain function \psis(\mu). Subject to a hypothesis on S, we obtain a partial description of the structure of such a random graph, including a condition for the existence (or not) of a giant component. The requisite hypothesis is in many cases benign, and applications are presented to a number of choices for the set S including the sets of (respectively) even and odd numbers. The random \emph{even} graph is related to the random-cluster model on the complete graph K_n. http://arxiv.org/abs/0712.0270 --------------------------------------------------------------- 6361. MEAN-FIELD BEHAVIOR FOR LONG- AND FINITE RANGE ISING MODEL, PERCOLATION AND SELF-AVOIDING WALK Markus Heydenreich and Remco van der Hofstad and Akira Sakai We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if $d>2(\alpha\wedge2)$ for self-avoiding walk and the Ising model, and $d>3(\alpha\wedge2)$ for percolation, where $d$ denotes the dimension and $\alpha$ the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (2007) http://arxiv.org/abs/0712.0312 --------------------------------------------------------------- 6362. DUALITY OF CHORDAL SLE Dapeng Zhan We prove that the outer boundary of the final hull of some chordal SLE$(\kappa;\vec{\rho})$ process has the same distribution as the image of some chordal SLE$(\kappa';\vec{\rho'})$ trace, where $\kappa>4$ and $\kappa'=16/\kappa$; and the reversal of some SLE$(4;\vec{\rho})$ trace has the same distribution as the time-change of some SLE$(4;\vec{\rho'})$ trace. And we also study some geometric properties of some chordal SLE$(\kappa;\vec {\rho})$ traces. http://arxiv.org/abs/0712.0332 --------------------------------------------------------------- 6363. RATES OF CONVERGENCE FOR MINIMAL DISTANCES IN THE CENTRAL LIMIT THEOREM UNDER PROJECTIVE CRITERIA J\'er\^ome Dedecker (LSTA) and Florence Merlev\`ede (PMA) and Emmanuel Rio (LM-Versailles) In this paper, we give estimates of ideal or minimal distances between the distribution of the normalized partial sum and the limiting Gaussian distribution for stationary martingale difference sequences or stationary sequences satisfying projective criteria. Applications to functions of linear processes and to functions of expanding maps of the interval are given. http://arxiv.org/abs/0712.0179 --------------------------------------------------------------- 6364. CONSTRAINED BSDE AND VISCOSITY SOLUTIONS OF VARIATION INEQUALITIES Shige Peng and Mingyu Xu In this paper, we study the relation between the smallest $g$- supersolution of constraint backward stochastic differential equation and viscosity solution of constraint semilineare parabolic PDE, i.e. variation inequalities. And we get an existence result of variation inequalities via constraint BSDE, and prove a uniqueness result under certain condition. http://arxiv.org/abs/0712.0306 --------------------------------------------------------------- 6365. LARGE DEVIATIONS FOR HEAVY-TAILED FACTOR MODELS Boualem Djehiche and Jens Svensson We study large deviation probabilities for a sum of dependent random variables from a heavy-tailed factor model, assuming that the components are regularly varying. We identify conditions where both the factor and the idiosyncratic terms contribute to the behaviour of the tail- probability of the sum. A simple conditional Monte Carlo algorithm is also provided together with a comparison between the simulations and the large deviation approximation. We also study large deviation probabilities for stochastic processes with factor structure. The processes involved are assumed to be Levy processes with regularly varying jump measures. Based on the results of the first part of the paper, we show that large deviations on a finite time interval are due to one large jump that can come from either the factor or the idiosyncratic part of the process. http://arxiv.org/abs/0712.0459 --------------------------------------------------------------- 6366. MULTIPLE EQUILIBRIA OF NONHOMOGENEOUS MARKOV CHAINS AND SELF- VALIDATING WEB RANKINGS Marianne Akian and Stephane Gaubert and Laure Ninove PageRank is a ranking of the web pages that measures how often a given web page is visited by a random surfer on the web graph, for a simple model of web surfing. It seems realistic that PageRank may also have an influence on the behavior of web surfers. We propose here a simple model taking into account the mutual influence between web ranking and web surfing. Our ranking, the T-PageRank, is a nonlinear generalization of the PageRank. It is defined as the limit, if it exists, of some nonlinear iterates. A positive parameter T, the temperature, measures the confidence of the web surfer in the web ranking. We prove that, when the temperature is large enough, the T-PageRank is unique and the iterates converge globally on the domain. But when the temperature is small, there may be several T-PageRanks, that may strongly depend on the initial ranking. Our analysis uses results of nonlinear Perron-Frobenius theory, Hilbert projective metric and Birkhoff's coefficient of ergodicity. http://arxiv.org/abs/0712.0469 --------------------------------------------------------------- 6367. GLAUBER DYNAMICS ON HYPERBOLIC GRAPHS: BOUNDARY CONDITIONS AND MIXING TIME Alessandra Bianchi We study a continuous time Glauber dynamics reversible with respect to the Ising model on hyperbolic graphs and analyze the effect of boundary conditions on the mixing time. Specifically, we consider the dynamics on an n- vertex ball of the hyperbolic graph $\H(v,s)$, where v is the number of neighbors of each vertex and s is the number of sides of each face, conditioned on having (+)-boundary. If v>4, s>3 and for all low enough temperatures (phase coexistence region) we prove that the spectral gap of this dynamics is bounded below by a constant independent of n. This implies that the mixing time grows at most linearly in n, in contrast to the free boundary case where it is polynomial with exponent growing with the inverse temperature $\b$. Such a result extends to hyperbolic graphs the work done by Martinelli, Sinclair and Weitz for the analogous system on regular tree graphs, and provides a further example of influence of the boundary condition on the mixing time. http://arxiv.org/abs/0712.0489 --------------------------------------------------------------- 6368. FRAGMENTING RANDOM PERMUTATIONS Christina Goldschmidt and James B. Martin and Dario Span\`o Problem 1.5.7 from Pitman's Saint-Flour lecture notes: Does there exist for each n a fragmentation process (\Pi_{n,k}, 1 \leq k \leq n) taking values in the space of partitions of {1,2,...,n} such that \Pi_{n,k} is distributed like the partition generated by cycles of a uniform random permutation of {1,2,...,n} conditioned to have k cycles? We show that the answer is yes. We also give a partial extension to general exchangeable Gibbs partitions. http://arxiv.org/abs/0712.0556 --------------------------------------------------------------- 6369. LARGE DEVIATIONS FOR LOCAL TIME FRACTIONAL BROWNIAN MOTION AND APPLICATIONS Mark M. Meerschaert and Erkan Nane and Yimin Xiao Let $W^H=\{W^H(t), t \in \rr\}$ be a fractional Brownian motion of Hurst index $H \in (0, 1)$ with values in $\rr$, and let $L = \{L_t, t \ge 0 \}$ be the local time process at zero of a strictly stable L\'evy process $X= \{X_t, t \ge 0\}$ of index $1<\alpha\leq 2$ independent of $W^H$. The $\a$- stable local time fractional Brownian motion $Z^H=\{Z^H(t), t \ge 0\}$ is defined by $Z^H(t) = W^H(L_t)$. The process $Z^H$ is self-similar with self-similarity index $H(1 - \frac 1 \alpha)$ and is related to the scaling limit of a continuous time random walk with heavy-tailed waiting times between jumps (\cite{coupleCTRW,limitCTRW}). However, $Z^H$ does not have stationary increments and is non-Gaussian. In this paper we establish large deviation results for the process $Z^H$. As applications we derive upper bounds for the uniform modulus of continuity and the laws of the iterated logarithm for $Z^H$. http://arxiv.org/abs/0712.0574 --------------------------------------------------------------- 6370. INVERSE PROBLEMS FOR REGULAR VARIATION OF LINEAR FILTERS, A CANCELLATION PROPERTY FOR $\SIGMA$-FINITE MEASURES, AND IDENTIFICATION OF STABLE LAWS Martin Jacobsen and Thomas Mikosch and Jan Rosinski and Gennady Samorodnitsky We study a group of related problems: the extent to which the presence of regular variation in the tail of certain $\sigma$-finite measures at the output of a linear filter determines the corresponding regular variation of a measure at the input to the filter. This turns out to be related to the presence of a particular cancellation property in $\sigma$-finite measures, which, in turn, is related to the uniqueness of the solution of certain functional equations. The techniques we develop are applied to weighted sums of iid random variables, to products of independent random variables, and to stochastic integrals with respect to L\'evy motions. http://arxiv.org/abs/0712.0576 --------------------------------------------------------------- 6371. SOME FAMILIES OF INCREASING PLANAR MAPS Marie Albenque (LIAFA) and Jean-Fran\c{c}ois Marckert (LaBRI) Stack-triangulations appear as natural objects when one wants to define some increasing families of triangulations by successive additions of faces. We investigate the asymptotic behavior of rooted stack-triangulations with $2n$ faces under two different distributions. We show that the uniform distribution on this set of maps converges, for a topology of local convergence, to a distribution on the set of infinite maps. In the other hand, we show that rescaled by $n^{1/2}$, they converge for the Gromov-Hausdorff topology on metric spaces to the continuum random tree introduced by Aldous. Under a distribution induced by a natural random construction, the distance between random points rescaled by $(6/11)\log n$ converge to 1 in probability. We obtain similar asymptotic results for a family of increasing quadrangulations. http://arxiv.org/abs/0712.0593 --------------------------------------------------------------- 6372. SEQUENTIAL TRACKING OF A HIDDEN MARKOV CHAIN USING POINT PROCESS OBSERVATIONS Erhan Bayraktar and Mike Ludkovski We study finite horizon optimal switching problems for hidden Markov chain models under partially observable Poisson processes. The controller possesses a finite range of strategies and attempts to track the state of the unobserved state variable using Bayesian updates over the discrete observations. Such a model has applications in economic policy making, staffing under variable demand levels and generalized Poisson disorder problems. We show regularity of the value function and explicitly characterize an optimal strategy. We also provide an efficient numerical scheme and illustrate our results with several computational examples. http://arxiv.org/abs/0712.0413 --------------------------------------------------------------- 6373. STOCHASTIC FITZHUGH-NAGUMO EQUATIONS ON NETWORKS WITH IMPULSIVE NOISE Stefano Bonaccorsi and Carlo Marinelli and Giacomo Ziglio We prove global well-posedness in the mild sense for a stochastic partial differential equation with a power-type nonlinearity and L\'evy noise. Equations of this type arise in models of neurophysiology. http://arxiv.org/abs/0712.0580 --------------------------------------------------------------- 6374. FILTRATIONS Delia Coculescu and Ashkan Nikeghbali In this article, we define the notion of a filtration and then give the basic theorems on initial and progressive enlargements of filtrations. http://arxiv.org/abs/0712.0622 --------------------------------------------------------------- 6375. CENTRAL LIMIT THEOREM FOR BRANCHING RANDOM WALKS IN RANDOM ENVIRONMENT Nobuo Yoshida We consider branching random walks in $d$-dimensional integer lattice with time-space i.i.d. offspring distributions. When $d \ge 3$ and the fluctuation of the environment is well moderated by the random walk, we prove a central limit theorem for the density of the population, together with upper bounds for the density of the most populated site and the replica overlap. We also discuss the phase transition of this model in connection with directed polymers in random environment. http://arxiv.org/abs/0712.0648 --------------------------------------------------------------- 6376. LOCALIZATION FOR BRANCHING RANDOM WALKS IN RANDOM ENVIRONMENT Yueyun Hu and Nobuo Yoshida We consider branching random walks in $d$-dimensional integer lattice with time-space i.i.d. offspring distributions. This model is known to exhibit a phase transition: If $d \ge 3$ and the environment is "not too random", then, the total population grows as fast as its expectation with strictly positive probability. If,on the other hand, $d \le 2$, or the environment is ``random enough", then the total population grows strictly slower than its expectation almost surely. We show the equivalence between the slow population growth and a natural localization property in terms of "replica overlap". We also prove a certain stronger localization property, whenever the total population grows strictly slower than its expectation almost surely. http://arxiv.org/abs/0712.0649 --------------------------------------------------------------- 6377. ERGODIC THEORY, ABELIAN GROUPS, AND POINT PROCESSES INDUCED BY STABLE RANDOM FIELDS Parthanil Roy We consider a point process sequence induced by a stationary symmetric alpha-stable (0 < alpha < 2) discrete parameter random field. It is easy to prove, following the arguments in the one-dimensional case in Resnick and Samorodnitsky (2004), that if the random field is generated by a dissipative group action then the point process sequence converges weakly to a cluster Poisson process. For the conservative case, no general result is known even in the one-dimensional case. We look at a specific class of stable random fields generated by conservative actions whose effective dimensions can be computed using the structure theorem of finitely generated abelian groups. The corresponding point processes sequence is not tight and hence needs to be properly normalized in order to ensure weak convergence. This weak limit is computed using extreme value theory and some counting techniques. http://arxiv.org/abs/0712.0688 --------------------------------------------------------------- 6378. SYSTEM RELIABILITY AND WEIGHTED LATTICE POLYNOMIALS Alexander Dukhovny and Jean-Luc Marichal The lifetime of a system of connected units under some natural assumptions can be represented as a random variable Y defined as a weighted lattice polynomial of random lifetimes of its components. As such, the concept of a random variable Y defined by a weighted lattice polynomial of (lattice-valued) random variables is considered in general and in some special cases. The central object of interest is the cumulative distribution function of Y. In particular, numerous results are obtained for lattice polynomials and weighted lattice polynomials in case of independent arguments and in general. For the general case, the technique consists in considering the joint probability generating function of "indicator" variables. A connection is studied between Y and order statistics of the set of arguments. http://arxiv.org/abs/0712.0707 --------------------------------------------------------------- 6379. ASYMPTOTICS FOR FIRST-PASSAGE TIMES OF L\'EVY PROCESSES AND RANDOM WALKS Denis Denisov and Vsevolod Shneer We study the exact asymptotics for the distribution of the first time $\tau_x$ a L\'evy process $X_t$ crosses a negative level $-x$. We prove that $\mathbf P(\tau_x>t)\sim V(x)\mathbf P(X_t\ge 0)/t$ as $t\to\infty$ for a certain function $V(x)$. Using known results for the large deviations of random walks we obtain asymptotics for $\mathbf P(\tau_x>t)$ explicitly in both light and heavy tailed cases. We also apply our results to find asymptotics for the distribution of the busy period in an M/G/1 queue. http://arxiv.org/abs/0712.0728 --------------------------------------------------------------- 6380. GLAUBER DYNAMICS FOR THE MEAN-FIELD ISING MODEL: CUT-OFF, CRITICAL POWER LAW, AND METASTABILITY David A. Levin and Malwina J. Luczak and Yuval Peres We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie-Weiss Model. For beta < 1, we prove that the dynamics exhibits a cut-off: the distance to stationarity drops from near 1 to near 0 in a window of order n centered at [2(1-beta)]^{-1} n log n. For beta = 1, we prove that the mixing time is of order n^{3/2}. For beta > 1, we study metastability. In particular, we show that the Glauber dynamics restricted to states of non-negative magnetization has mixing time O(n log n). http://arxiv.org/abs/0712.0790 --------------------------------------------------------------- 6381. REFLECTED BROWNIAN MOTION IN A WEDGE: SUM-OF-EXPONENTIAL STATIONARY DENSITIES A. B. Dieker and J. Moriarty We give necessary and sufficient conditions for the stationary density of reflected Brownian motion (RBM) in a wedge to be written as a finite sum of terms of exponential product form. Relying on geometric ideas reminiscent of the reflection principle, we give an explicit formula for the density in such cases. We also show that the density can be written as a determinant. http://arxiv.org/abs/0712.0844 --------------------------------------------------------------- 6382. HUA-PICKRELL MEASURES ON GENERAL COMPACT GROUPS Paul Bourgade and Ashkan Nikeghbali and Alain Rouault Take a generic subgroup $\mathcal{G}$, endowed with its Haar measure, from $U(n,K)$, the unitary group of dimension $n$ over the field $K$ of real, complex or quaternion numbers. We give some equalities in law for $Z:=\det(\Id-G)$, $G\in\mathcal{G}$ : under some general conditions, $Z$ can be decomposed as a product of independent random variables, whose laws are explicitly known (Section 2). Consequently $\mathcal{G}$, endowed with a generalization of its Haar measure (the Hua-Pickrell measure), can be generated as a product of independent reflections. This constitutes a generalization of the well known Ewens sampling formula, corresponding to $\mathcal{G}=\mathcal{S}_n$, the $n$-dimensional symmetric group. Eventually, explicit determinantal point processes can be associated to the spectrum induced by the Hua-Pickrell measures, implying asymptotics on correlation functions. http://arxiv.org/abs/0712.0848 --------------------------------------------------------------- 6383. WHAT IS THE DIFFERENCE BETWEEN A SQUARE AND A TRIANGLE? V. Limic and P. Tarres We offer a reader-friendly introduction to the attracting edge problem (also known as the "triangle conjecture") and its most general current solution of Limic and Tarr\`es (2007). Little original research is reported; rather this article ``zooms in'' to describe the essential characteristics of two different techniques/approaches verifying the almost sure existence of the attracting edge for the strongly edge reinforced random walk (SERRW) on a square. Both arguments extend straightforwardly to the SERRW on even cycles. Finally, we show that the case where the underlying graph is a triangle cannot be studied by a simple modification of either of the two techniques. http://arxiv.org/abs/0712.0958 --------------------------------------------------------------- 6384. HIGH RESOLUTION QUANTIZATION AND ENTROPY CODING OF JUMP PROCESSES Frank Aurzada and Steffen Dereich and Michael Scheutzow and Christian Vormoor We study the quantization problem for certain types of jump processes. The probabilities for the number of jumps are assumed to be bounded by Poisson weights. Otherwise, jump positions and increments can be rather generally distributed and correlated. We show in particular that in many cases entropy coding error and quantization error have distinct rates. Finally, we investigate the quantization problem for the special case of $\mathbb{R}^d$-valued compound Poisson processes. http://arxiv.org/abs/0712.0964 --------------------------------------------------------------- 6385. ON CONTINUOUS STATE BRANCHING PROCESSES: CONDITIONING AND SELF-SIMILARITY A.E. Kyprianou and J.C. Pardo In this paper, for $\alpha\in (1, 2}$ we show that the $\alpha$- stable continuous-state branching process and the associated process conditioned never to become extinct are positive self-similar Markov processes. Understanding the interaction of the Lamperti transformation for continuous-state branching processes and the Lamperti transformation for positive self-similar Markov processes permits accessto a number of explicit results concerning the paths of stable-continuous branching processes and its conditioned version. http://arxiv.org/abs/0712.0987 --------------------------------------------------------------- 6386. THE DERIVATIVES OF ASIAN CALL OPTION PRICES Jungmin Choi and Kyounghee Kim The distribution of a time integral of geometric Brownian motion is not well understood. To price an Asian option and to obtain measures of its dependence on the parameters of time, strike price, and underlying market price, it is essential to have the distribution of time integral of geometric Brownian motion and it is also required to have a way to manipulate its distribution. We present integral forms for key quantities in the price of Asian option and its derivatives ({\it{delta, gamma,theta, and vega}}). For example for any $a>0$ $\mathbb{E} [ (A_t -a)^+] = t -a + a^{2} \mathbb{E} [ (a+A_t)^{-1} \exp (\frac{2M_t}{a+ A_t} - \frac{2}{a}) ]$, where $A_t = \int^t_0 \exp (B_s -s/2) ds$ and $M_t =\exp (B_t -t/2).$ http://arxiv.org/abs/0712.1093 --------------------------------------------------------------- 6387. LAW OF THE EXPONENTIAL FUNCTIONAL OF A NEW FAMILY OF ONE-SIDED LEVY PROCESSES VIA SELF-SIMILAR CONTINUOUS STATE BRANCHING PROCESSES WITH IMMIGRATION AND THE WRIGHT HYPERGEOMETRIC FUNCTIONS P. Patie We first introduce and derive some basic properties of a two- parameters family of one-sided Levy processes. Their Laplace exponents are given in terms of the Pochhammer symbol. This family includes, in a limit case, the family of Brownian motion with drifts. Then, we proceed by computing the density of the law of the exponential functional associated to some elements of this family (and their dual) and some transformations of these elements. These densities are expressed in terms of the Wright hypergeometric functions. By means of probabilistic arguments, we derive some interesting properties enjoyed by these functions. On the way we also characterize explicitly the density of the semi-groups of the family of self-similar continuous state branching processes with immigration. http://arxiv.org/abs/0712.1115 --------------------------------------------------------------- 6388. INTRODUCTION TO (GENERALIZED) GIBBS MEASURES Arnaud Le Ny These notes have been written to complete a mini-course "Introduction to (generalized) Gibbs measures" given at the universities UFMG (Universidade Federal de Minas Gerais, Belo Horizonte, Brasil) and UFRGS (Universidade Federal do Rio Grande do Sul, Porto Alegre, Brasil) during the first semester 2007. The main goal of the lectures was to describe Gibbs and generalized Gibbs measures on lattices at a rigorous mathematical level, as equilibirum states of systems of a huge number of particles in interaction. In particular, our main message is that although the historical approach based on potentials has been rather successful from a physical point of view, one has to insist on (almost sure) continuity properties of conditional probabilities to get a proper mathematical framework. http://arxiv.org/abs/0712.1171 --------------------------------------------------------------- 6389. AIRY KERNEL WITH TWO SETS OF PARAMETERS IN DIRECTED PERCOLATION AND RANDOM MATRIX THEORY A. Borodin; S. Peche We introduce a generalization of the extended Airy kernel with two sets of real parameters. We show that this kernel arises in the edge scaling limit of correlation kernels of determinantal processes related to a directed percolation model and to an ensemble of random matrices. http://arxiv.org/abs/0712.1086 --------------------------------------------------------------- 6390. CONTINUOUS-TIME TRADING AND EMERGENCE OF RANDOMNESS, I Vladimir Vovk A new definition of events of game-theoretic probability zero in continuous time is proposed and used to prove results suggesting that trading in financial markets results in the emergence of properties usually associated with randomness. This paper concentrates on "qualitative" results, stated in terms of order (or order topology) rather than in terms of the precise values taken by the price processes. http://arxiv.org/abs/0712.1275 --------------------------------------------------------------- 6391. AN HILBERT SPACE APPROACH FOR A CLASS OF ARBITRAGE FREE IMPLIED VOLATILITIES MODELS G. Fabbri and B. Goldys We present an Hilbert space formulation for a set of implied volatility models introduced in \cite{BraceGoldys01} in which the authors studied conditions for a family of European call options, varying the maturing time and the strike price $T$ an $K$, to be arbitrage free. The arbitrage free conditions give a system of stochastic PDEs for the evolution of the implied volatility surface ${\hat\sigma}_t(T,K)$. We will focus on the family obtained fixing a strike $K$ and varying $T$. In order to give conditions to prove an existence-and-uniqueness result for the solution of the system it is here expressed in terms of the square root of the forward implied volatility and rewritten in an Hilbert space setting. The existence and the uniqueness for the (arbitrage free) evolution of the forward implied volatility, and then of the the implied volatility, among a class of models, are proved. Specific examples are also given. http://arxiv.org/abs/0712.1343 --------------------------------------------------------------- 6392. COMMUTATION RELATIONS AND MARKOV CHAINS Jason Fulman It is shown that the combinatorics of commutation relations is well suited for analyzing the convergence rate of certain Markov chains. Examples studied include random walk on irreducible representations, a local random walk on partitions whose stationary distribution is the Ewens distribution, and some birth-death chains. http://arxiv.org/abs/0712.1375 --------------------------------------------------------------- 6393. LYAPUNOV EXPONENTS OF FREE OPERATORS Vladislav Kargin Lyapunov exponents of a dynamical system are a useful tool to gauge the stability and complexity of the system. This paper offers a definition of Lyapunov exponents for a sequence of free linear operators. The definition is based on the concept of the extended Fuglede-Kadison determinant. We establish the existence of Lyapunov exponents, derive formulas for their calculation, and show that Lyapunov exponents are additive with respect to operator product. We illustrate these results using an example of free operators whose singular values are distributed by the Marchenko-Pastur law, and relate this example to C. M. Newman's "triangle" law for the distribution of Lyapunov exponents of large random matrices with independent Gaussian entries. As an interesting by-product of our results, we derive a relation between the extended Fuglede-Kadison determinant and Voiculescu's S-transform, which sheds some light on the multiplicativity of the S-transform. http://arxiv.org/abs/0712.1378 --------------------------------------------------------------- 6394. CONTINUOUS-TIME TRADING AND EMERGENCE OF RANDOMNESS, II Vladimir Vovk This paper continues investigation of randomness-type properties emerging in idealized financial markets with continuous price processes. It is shown that the strong variation exponent of non-constant price processes has to be 2, as in the case of Brownian motion. http://arxiv.org/abs/0712.1483 --------------------------------------------------------------- 6395. HOW UNIVERSAL ARE ASYMPTOTICS OF DISCONNECTION TIMES IN DISCRETE CYLINDERS? Alain-Sol Sznitman We investigate the disconnection time of a simple random walk in a discrete cylinder with a large finite connected base. In a recent article of A. Dembo and the author it was found that for large $N$ the disconnection time of $G_N\times\mathbb{Z}$ has rough order $|G_N|^2$, when $G_N=(\mathbb{Z}/N\mathbb{Z})^d$. In agreement with a conjecture by I. Benjamini, we show here that this behavior has broad generality when the bases of the discrete cylinders are large connected graphs of uniformly bounded degree. http://arxiv.org/abs/0712.1497 --------------------------------------------------------------- 6396. RATNER'S THEOREM ON HOROCYCLIC FLOWS John H. Hubbard and Robyn L. Miller We provide a self-contained, accessible introduction to Ratner's Equidistribution Theorem in the special case of horocyclic flow on a complete hyperbolic surface of finite area. We also prove a result due to Breuillard: on the modular surface an arbitrary uncentered random walk on the horocycle through almost any point will fail to equidistribute, even though the horocycles are themselves equidistributed. http://arxiv.org/abs/0712.1300 --------------------------------------------------------------- 6397. SPECTRUM OF THE PRODUCT OF TOEPLITZ MATRICES WITH APPLICATION IN PROBABILITY Bernard Bercu and Jean-Francois Bony and Vincent Bruneau We study the spectrum of the product of two Toeplitz operators. Assume that the symbols of these operators are continuous and real-valued and that one of them is non-negative. We prove that the spectrum of the product of finite section Toeplitz matrices converges to the spectrum of the product of the semi-infinite Toeplitz operators. We give an example showing that the supremum of this set is not always the supremum of the product of the two symbols. Finally, we provide an application in probability which is the first motivation of this study. More precisely, we obtain a large deviation principle for Gaussian quadratic forms. http://arxiv.org/abs/0712.1302 --------------------------------------------------------------- 6398. NON-INTERSECTING SQUARED BESSEL PATHS AND MULTIPLE ORTHOGONAL POLYNOMIALS FOR MODIFIED BESSEL WEIGHTS A.B.J. Kuijlaars and A. Martinez-Finkelshtein and and F. Wielonsky We study a model of $n$ non-intersecting squared Bessel processes in the confluent case: all paths start at time $t = 0$ at the same positive value $x = a$, remain positive, and are conditioned to end at time $t = T$ at $x = 0$. In the limit $n \to \infty$, after appropriate rescaling, the paths fill out a region in the $tx$-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at $x = 0$, but at a certain critical time $t^*$ the smallest paths hit the hard edge and from then on are stuck to it. For $t \neq t^*$ we obtain the usual scaling limits from random matrix theory, namely the sine, Airy, and Bessel kernels. A key fact is that the positions of the paths at any time $t$ constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel- type weights. As a consequence, there is a $3 \times 3$ matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large $n$ limit using the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest. http://arxiv.org/abs/0712.1333 --------------------------------------------------------------- 6399. MEASURES ON TWO-COMPONENT CONFIGURATION SPACES D.L. Finkelshtein We study measures on the configuration spaces of two type particles. Gibbs measures on the such spaces are described. Main properties of corresponding relative energies densities and correlation functions are considered. In particular, we show that a support set for the such Gibbs measure is the set of pairs of non-intersected configurations. http://arxiv.org/abs/0712.1401 --------------------------------------------------------------- 6400. THE JOINT DISTRIBUTION OF OCCUPATION TIMES OF SKIP-FREE MARKOV PROCESSES AND A CLASS OF MULTIVARIATE EXPONENTIAL DISTRIBUTIONS Kshitij Khare For a skip-free Markov process on non-negative integers with generator matrix Q, we evaluate the joint Laplace transform of the occupation times before hitting the state n (starting at 0). This Laplace transform has a very straightforward and familiar expression. We investigate the properties of this Laplace transform, especially the conditions under which the occupation times form a Markov chain. http://arxiv.org/abs/0712.1646 --------------------------------------------------------------- 6401. LOCAL TAIL BOUNDS FOR FUNCTIONS OF INDEPENDENT RANDOM VARIABLES Luc Devroye and G\'abor Lugosi It is shown that functions defined on $\{0,1,...,r-1\}^n$ satisfying certain conditions of bounded differences that guarantee sub-Gaussian tail behavior also satisfy a much stronger ``local'' sub-Gaussian property. For self-bounding and configuration functions we derive analogous locally subexponential behavior. The key tool is Talagrand's [Ann. Probab. 22 (1994) 1576--1587] variance inequality for functions defined on the binary hypercube which we extend to functions of uniformly distributed random variables defined on $\{0,1,...,r-1\}^n$ for $r\ge2$. http://arxiv.org/abs/0712.1686 --------------------------------------------------------------- 6402. RANDOM GRAPH MODELS OF COMMUNICATION NETWORK TOPOLOGIES Hannu Reittu and Ilkka Norros We consider a variant of so called power-law random graph. A sequence of expected degrees corresponds to a power-law degree distribution with finite mean and infinite variance. In previous works the asymptotic picture with number of nodes limiting to infinity has been considered. It was found that an interesting structure appears. It has resemblance with such graphs like the Internet graph. Some simulations have shown that a finite sized variant has similar properties as well. Here we investigate this case in more analytical fashion, and, with help of some simple lower bounds for large valued expectations of relevant random variables, we can shed some light into this issue. A new term, 'communication range random graph' is introduced to emphasize that some further restrictions are needed to have a relevant random graph model for a reasonable sized communication network, like the Internet. In this case a pleasant model is obtained, giving the opportunity to understand such networks on an intuitive level. This would be beneficial in order to understand, say, how a particular routing works in such networks. http://arxiv.org/abs/0712.1690 --------------------------------------------------------------- 6403. RANDOM CLUSTER TESSELLATIONS Kai Matzutt This article describes, in elementary terms, a generic approach to produce discrete random tilings and similar random structures by using point process theory. The standard Voronoi and Delone tilings can be constructed in this way. For this purpose, convex polytopes are replaced by their vertex sets. Three explicit constructions are given to illustrate the concept. http://arxiv.org/abs/0712.1684 --------------------------------------------------------------- 6404. POISSON MATCHING Alexander E. Holroyd and Robin Pemantle and Yuval Peres and Oded Schramm Suppose that red and blue points occur as independent homogeneous Poisson processes in R^d. We investigate translation-invariant schemes for perfectly matching the red points to the blue points. For any such scheme in dimensions d=1,2, the matching distance X from a typical point to its partner must have infinite d/2-th moment, while in dimensions d>=3 there exist schemes where X has finite exponential moments. The Gale-Shapley stable marriage is one natural matching scheme, obtained by iteratively matching mutually closest pairs. A principal result of this paper is a power law upper bound on the matching distance $X$ for this scheme. A power law lower bound holds also. In particular, stable marriage is close to optimal (in tail behavior) in d=1, but far from optimal in d>=3. For the problem of matching Poisson points of a single color to each other, in d=1 there exist schemes where X has finite exponential moments, but if we insist that the matching is a deterministic factor of the point process then X must have infinite mean. http://arxiv.org/abs/0712.1867 --------------------------------------------------------------- 6405. SUPERCRITICAL GENERAL BRANCHING PROCESSES CONDITIONED ON EXTINCTION ARE SUBCRITICAL Peter Jagers and Andreas Nordvall Lager{\aa}s It is well known that a simple, supercritical Bienaym\'e-Galton- Watson process turns into a subcritical such process, if conditioned to die out. We prove that the corresponding holds true for general, multi-type branching, where child-bearing may occur at different ages, life span may depend upon reproduction, and the whole course of events is thus affected by conditioning upon extinction. http://arxiv.org/abs/0712.1872 --------------------------------------------------------------- 6406. CYCLES OF RANDOM PERMUTATIONS WITH RESTRICTED CYCLE LENGTHS Florent Benaych-Georges (PMA) We prove some general results about the asymptotics of the distribution of the number of cycles of given length of a random permutation which distribution is invariant under conjugation. These results were first established to be applied in a forthcoming paper (Cycles of free words in several random permutations with restricted cycles lengths) were we prove results about cycles of random permutations which can be written as free words in several independent random permutations. However, we also apply them here to prove asymptotic results about random permutations with restricted cycle lengths. More specifically, for $A$ set of positive integers, we consider a random permutation chosen uniformly among permutations of $\{1,..., n\}$ which have all their cycle lengths in $A$, and then let $n$ tend to infinity. We prove that if $A$ is infinite and large enough, then the number of cycles of different given cycle lengths of this random permutation are asymptotically independent and distributed according to Poisson distributions. In the case where $A$ is finite, we prove that the behavior of these random variables is completely different: cycles with length $\max A$ are predominant. http://arxiv.org/abs/0712.1903 --------------------------------------------------------------- 6407. PROOFS OF THE MARTINGALE FCLT Ward Whitt This is an expository review paper elaborating on the proof of the martingale functional central limit theorem (FCLT). This paper also reviews tightness and stochastic boundedness, highlighting one-dimensional criteria for tightness used in the proof of the martingale FCLT. This paper supplements the expository review paper Pang, Talreja and Whitt (2007) illustrating the ``martingale method'' for proving many-server heavy-traffic stochastic-process limits for queueing models, supporting diffusion-process approximations. http://arxiv.org/abs/0712.1929 --------------------------------------------------------------- 6408. FACILITATED SPIN MODELS: RECENT AND NEW RESULTS Nicoletta Cancrini and Fabio Martinelli and Cyril Roberto and Cristina Toninelli Facilitated or kinetically constrained spin models (KCSM) are a class of interacting particle systems reversible w.r.t. to a simple product measure. Each dynamical variable (spin) is re-sampled from its equilibrium distribution only if the surrounding configuration fulfills a simple local constraint which \emph{does not involve} the chosen variable itself. Such simple models are quite popular in the glass community since they display some of the peculiar features of glassy dynamics, in particular they can undergo a dynamical arrest reminiscent of the liquid/glass transitiom. Due to the fact that the jumps rates of the Markov process can be zero, the whole analysis of the long time behavior becomes quite delicate and, until recently, KCSM have escaped a rigorous analysis with the notable exception of the East model. In these notes we will mainly review several recent mathematical results which, besides being applicable to a wide class of KCSM, have contributed to settle some debated questions arising in numerical simulations made by physicists. We will also provide some interesting new extensions. In particular we will show how to deal with interacting models reversible w.r.t. to a high temperature Gibbs measure and we will provide a detailed analysis of the so called one spin facilitated model on a general connected graph. http://arxiv.org/abs/0712.1934 --------------------------------------------------------------- 6409. SCALING LIMIT AND AGING FOR DIRECTED TRAP MODELS Olivier Zindy (WIAS) We consider one-dimensional directed trap models and suppose that the trapping times are heavy-tailed. We obtain the inverse of a stable subordinator as scaling limit and prove an aging phenomenon expressed in terms of the generalized arcsine law. These results confirm the status of universality described by Ben Arous and \v{C}ern\'y for a large class of graphs. http://arxiv.org/abs/0712.1951 --------------------------------------------------------------- 6410. CONTINUUM LIMITS OF RANDOM MATRICES AND THE BROWNIAN CAROUSEL Benedek Valko and Balint Virag We show that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general beta siblings converge to Sine_beta, a translation invariant point process. This process has a geometric description in term of the Brownian carousel, a deterministic function of Brownian motion in the hyperbolic plane. The Brownian carousel, a description of the a continuum limit of random matrices, provides a convenient way to analyze the limiting point processes. We show that the gap probability of Sine_beta is continuous in the gap size and $\beta$, and compute its asymptotics for large gaps. Moreover, the stochastic differential equation version of the Brownian carousel exhibits a phase transition at beta=2. http://arxiv.org/abs/0712.2000 --------------------------------------------------------------- 6411. CRITIQUE DU RAPPORT SIGNAL \`A BRUIT EN TH\'EORIE DE L'INFORMATION -- A CRITICAL APPRAISAL OF THE SIGNAL TO NOISE RATIO IN INFORMATION THEORY Michel Fliess (INRIA Futurs) The signal to noise ratio, which plays such an important role in information theory, is shown to become pointless in digital communications where - symbols are modulating carriers, which are solutions of linear differential equations with polynomial coefficients, - demodulations is achieved thanks to new algebraic estimation techniques. Operational calculus, differential algebra and nonstandard analysis are the main mathematical tools. http://arxiv.org/abs/0712.1875 --------------------------------------------------------------- 6412. ASYMPTOTIC DISTRIBUTIONS AND CHAOS FOR THE SUPERMARKET MODEL Malwina J. Luczak and Colin McDiarmid In the supermarket model there are n queues, each with a unit rate server. Customers arrive in a Poisson process at rate \lambda n, where 0< \lambda <1. Each customer chooses d > 2 queues uniformly at random, and joins a shortest one. It is known that the equilibrium distribution of a typical queue length converges to a certain explicit limiting distribution as n -> oo. We quantify the rate of convergence by showing that the total variation distance between the equilibrium distribution and the limiting distribution is essentially of order n^{-1}; and we give a corresponding result for systems starting from quite general initial conditions (not in equilibrium). Further, we quantify the result that the systems exhibit chaotic behaviour: we show that the total variation distance between the joint law of a fixed set of queue lengths and the corresponding product law is essentially of order at most n^{-1}. http://arxiv.org/abs/0712.2091 --------------------------------------------------------------- 6413. REPRESENTATION THEOREMS FOR BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL EQUATIONS Auguste Aman (LMAI) In this paper we study the class of backward doubly stochastic differential equation (BDSDE, for short) whose terminal value depends on the history of forward diffusion. We first establish a probabilistic representation for the spatial gradient of the stochastic viscosity solution to a quasilinear parabolic SPDE in the spirit of the Feynman-Kac formula, without using the derivatives of the coefficients of the corresponding BDSDE. Then such a representation leads to a closed-form representation of the martingale integrand of BDSDE, under only standard Lipschitz condition on the coefficients. http://arxiv.org/abs/0712.2219 --------------------------------------------------------------- 6414. LARGE DEVIATIONS FOR RANDOM TREES Yuri Bakhtin and Christine Heitsch We consider large random trees under Gibbs distributions and prove a Large Deviation Principle (LDP) for the distribution of degrees of vertices of the tree. The LDP rate function is given explicitly. An immediate consequence is a Law of Large Numbers for the distribution of vertex degrees in a large random tree. Our motivation for this study comes from the analysis of RNA secondary structures. http://arxiv.org/abs/0712.2253 --------------------------------------------------------------- 6415. COMPETING PARTICLE SYSTEMS AND THE GHIRLANDA-GUERRA IDENTITIES Louis-Pierre Arguin We study point processes on the real line whose configurations X can be ordered decreasingly and evolve by increments which are functions of correlated gaussian variables. The correlations are intrinsic to the points and quantified by a matrix Q={q_ij}. Quasi-stationary systems are those for which the law of (X,Q) is invariant under the evolution up to translation of X. It was conjectured by Aizenman and co-authors that the matrix Q of robustly quasi-stationary systems must exhibit a hierarchal structure. This was established recently, up to a natural decomposition of the system, whenever the set S_Q of values assumed by q_ij is finite. In this paper, we study the general case where S_Q may be infinite. Using the past increments of the evolution, we show that the law of robustly quasi-stationary systems must obey the Ghirlanda-Guerra identities, which first appear in the study of spin glass models. This provides strong evidence that the above conjecture also holds in the general case. http://arxiv.org/abs/0712.2338 --------------------------------------------------------------- 6416. PARTICLE APPROXIMATION OF THE WASSERSTEIN DIFFUSION Sebastian Andres and Max-K. von Renesse We construct a system of interacting two-sided Bessel processes on the unit interval and show that the associated empirical measure process converges to the Wasserstein Diffusion, assuming that Markov uniqueness holds for the generating Wasserstein Dirichlet form. The proof is based on the variational convergence of an associated sequence of Dirichlet forms in the generalized Mosco sense of Kuwae and Shioya. http://arxiv.org/abs/0712.2387 --------------------------------------------------------------- 6417. LARGE DEVIATIONS FOR RIESZ POTENTIALS OF ADDITIVE PROCESSES R. Bass and X. Chen and J. Rosen We study functionals of the form \[\zeta_{t}=\int_0^{t}...\int_0^ {t} | X_1(s_1)+...+ X_p(s_p)|^{-\sigma}ds_1... ds_p\] where $X_1(t),..., X_p (t)$ are i.i.d. $d$-dimensional symmetric stable processes of index $0<\bb\le 2 $. We obtain results about the large deviations and laws of the iterated logarithm for $\zeta_{t}$. http://arxiv.org/abs/0712.2401 --------------------------------------------------------------- 6418. LIMITS OF ONE DIMENSIONAL DIFFUSIONS George Lowther In this paper we look at the properties of limits of a sequence of real valued time inhomogeneous diffusions. When convergence is only in the sense of finite-dimensional distributions then the limit does not have to be a diffusion. However, we show that as long as the drift terms satisfy a Lipschitz condition and the limit is continuous in probability, then it will lie in a class of processes that we refer to as almost-continuous diffusions. These processes are strong Markov and satisfy an `almost-continuity' condition. We also give a simple condition for the limit to be a continuous diffusion. These results contrast with the multidimensional case where, as we show with an example, a sequence of two dimensional martingale diffusions can converge to a process that is both discontinuous and non-Markov. http://arxiv.org/abs/0712.2428 --------------------------------------------------------------- 6419. LIMITS TO CONSISTENT ON-LINE FORECASTING FOR ERGODIC TIME SERIES L. Gyorfi and G. Morvai and and S. Yakowitz This study concerns problems of time-series forecasting under the weakest of assumptions. Related results are surveyed and are points of departure for the developments here, some of which are new and others are new derivations of previous findings. The contributions in this study are all negative, showing that various plausible prediction problems are unsolvable, or in other cases, are not solvable by predictors which are known to be consistent when mixing conditions hold. http://arxiv.org/abs/0712.2430 --------------------------------------------------------------- 6420. NUMERICAL SENSITIVITY AND EFFICIENCY IN THE TREATMENT OF EPISTEMIC AND ALEATORY UNCERTAINTY Eric Chojnacki (IRSN) and Jean Baccou (IRSN) and S\'ebastien Destercke (IRSN, IRIT) The treatment of both aleatory and epistemic uncertainty by recent methods often requires an high computational effort. In this abstract, we propose a numerical sampling method allowing to lighten the computational burden of treating the information by means of so-called fuzzy random variables. http://arxiv.org/abs/0712.2141 --------------------------------------------------------------- 6421. TAKACS' ASYMPTOTIC THEOREM AND ITS APPLICATIONS: A SURVEY Vyacheslav M. Abramov The book of Lajos Tak\'acs \emph{Combinatorial Methods in the Theory of Stochastic Processes} has been published in 1967. It discusses various problems associated with $$ P_{k,i}=\mathrm{P}\left\{\sup_{1\leq n\leq\rho(i)}(N_n-n)0$, and $\rho(i)$ is the smallest $n$ such that $N_n=n-i$, $i \geq1$. (If there is no such $n$, then $\rho(i)=\infty$.) (*) is a discrete generalization of the classic ruin probability, and its value is represented as $P_{k,i}={Q_{k-i}}/{Q_k}$, where the sequence $\{Q_k\}_{k\geq0}$ satisfies the recurrence relation of convolution type: $Q_0\neq0$ and $Q_k=\sum_{j=0}^k\pi_jQ_{k-j+1}$. Since 1967 there have been many papers related to applications of the generalized classic ruin probability. The present survey concerns only with one of the areas of application associated with asymptotic behavior of $Q_k$ as $k\to\infty$. The theorem on asymptotic behaviour of $Q_k$ as $k\to \infty$ and further properties of that limiting sequence are given on pages 22-23 of the aforementioned book by Tak\'acs. In the present survey we discuss applications of Tak\'acs' asymptotic theorem and other related results in queueing theory, telecommunication systems and dams. Most of the results of this survey are based on the work of the author and have appeared during the last years. http://arxiv.org/abs/0712.2480 --------------------------------------------------------------- 6422. FRACTIONAL MOMENT BOUNDS AND DISORDER RELEVANCE FOR PINNING MODELS B. Derrida and G. Giacomin and H. Lacoin and F. L. Toninelli We study the critical point of directed pinning/wetting models with quenched disorder. The distribution K(.) of the location of the first contact of the (free) polymer with the defect line is assumed to be of the form K(n)=n^{-\alpha-1}L(n), with L(.) slowly varying. The model undergoes a (de)-localization phase transition: the free energy (per unit length) is zero in the delocalized phase and positive in the localized phase. For \alpha<1/2 it is known that disorder is irrelevant: quenched and annealed critical points coincide for small disorder, as well as quenched and annealed critical exponents. The same has been proven also for \alpha=1/2, but under the assumption that L(.) diverges sufficiently fast at infinity, an hypothesis that is not satisfied in the (1+1)-dimensional wetting model considered by Forgacs et al. (1986) and Derrida et al. (1992), where L(.) is asymptotically constant. Here we prove that, if 1/2<\alpha<1 or \alpha >1, then quenched and annealed critical points differ whenever disorder is present, and we give the scaling form of their difference for small disorder. In agreement with the so- called Harris criterion, disorder is therefore relevant in this case. In the marginal case \alpha=1/2, under the assumption that L(.) vanishes sufficiently fast at infinity, we prove that the difference between quenched and annealed critical points, which is known to be smaller than any power of the disorder strength, is positive: disorder is marginally relevant. Again, the case considered by Forgacs et al. (1986) and Derrida et al. (1992) is out of our analysis and remains open. http://arxiv.org/abs/0712.2515 --------------------------------------------------------------- 6423. WHEN DO STOCHASTIC MAX-PLUS LINEAR SYSTEMS HAVE A CYCLE TIME ? Glenn Merlet (LIAFA) We analyze the asymptotic behavior of the sequence of random variables (x(n, x0))n \in N defined by x(0, x0) = x0 and x(n+1, x0) = A(n)x(n, x0), where (A(n))n \in N is a stationary and ergodic sequence of random matrices with entries in the semiring (R \cup {-\infinity}, max, +). Such sequences model a large class of discrete event systems, among which timed event graphs, 1-bounded Petri nets, some queuing networks, train or computer networks. We give a necessary condition for 1/n x(n, x0) n \in N to converge almost-surely, which proves to be sufficient when the A(n) are i.i.d. Moreover, we construct a new example, in which (A(n))n \in N is strongly mixing, that condition is satisfied, but 1/n x(n, x0) n \in N do not converge almost-surely. http://arxiv.org/abs/0712.2559 --------------------------------------------------------------- 6424. CONVEX ENTROPY DECAY VIA THE BOCHNER-BAKRY-EMERY APPROACH Pietro Caputo and Paolo Dai Pra and Gustavo Posta We develop a method, based on a Bochner-type identity, to obtain estimates on the exponential rate of decay of the relative entropy from equilibrium of Markov processes in discrete settings. When this method applies the relative entropy decays in a convex way. The method is shown to be rather powerful when applied to a class of birth and death processes. We then consider other examples, including inhomogeneous zero-range processes and Bernoulli- Laplace models. For these two models, known results were limited to the homogeneous case, and obtained via the martingale approach, whose applicability to inhomogeneous models is still unclear. http://arxiv.org/abs/0712.2578 --------------------------------------------------------------- 6425. MINIMA IN BRANCHING RANDOM WALKS L. Addario-Berry and B.A. Reed Given a branching random walk, let M_n be the minimum position of any member of the n'th generation. We calculate the expected value of M_n to within O(1) and prove exponential tail bounds for M_n around its expected value, under quite general conditions on the branching random walk. In particular, together with work of Bramson (1978), our results fully characterize the possible behavior of M_n when the branching random walk has bounded branching and step size. http://arxiv.org/abs/0712.2582 --------------------------------------------------------------- 6426. STRONGLY CONSISTENT NONPARAMETRIC FORECASTING AND REGRESSION FOR STATIONARY ERGODIC SEQUENCES S. Yakowitz and L. Gyorfi and J. Kieffer and G. Morvai Let $\{(X_i,Y_i)\}$ be a stationary ergodic time series with $(X,Y) $ values in the product space $\R^d\bigotimes \R .$ This study offers what is believed to be the first strongly consistent (with respect to pointwise, least- squares, and uniform distance) algorithm for inferring $m(x)=E[Y_0|X_0=x]$ under the presumption that $m(x)$ is uniformly Lipschitz continuous. Auto- regression, or forecasting, is an important special case, and as such our work extends the literature of nonparametric, nonlinear forecasting by circumventing customary mixing assumptions. The work is motivated by a time series model in stochastic finance and by perspectives of its contribution to the issues of universal time series estimation. http://arxiv.org/abs/0712.2592 --------------------------------------------------------------- 6427. RATE OF RELAXATION FOR A MEAN-FIELD ZERO-RANGE PROCESS B.T. Graham We introduce a mean-field zero-range process. It is a Markov chain model for a microcanonical ensemble. We prove that the process converges to a fluid limit. The fluid limit rapidly relaxes to the appropriate Gibbs distribution. http://arxiv.org/abs/0712.2599 --------------------------------------------------------------- 6428. RANDOM WALKS AND NON-OVERSHOOTING LEVY PROCESSES Sergey G. Foss and Anatolii A. Puhalskii Let $\xi_1,\xi_2,...$ be i.i.d. random variables with negative mean. Suppose that $\mathbf{E}\exp(\lambda\xi_1)<\infty$ for some $\lambda>0$ and that there exists $\gamma>0$ with $\mathbf{E}\exp(\gamma\xi_1)=1$ . It is known that if, in addition, $\mathbf{E} \xi_1\exp(\gamma\xi_1)<\infty$, then the most likely way for the random walk $S_k=\sum_{i=1}^k\xi_i$ to reach a high level is to follow a straight line with a positive slope. We study the case where $\mathbf{E} \xi_1\exp(\gamma\xi_1)=\infty$. Assuming that the distribution $\exp(\gamma x) \mathbf{P}(\xi_1\in dx) $ belongs to the domain of attraction of a spectrally positive stable law, we obtain a weak convergence limit theorem as $r\to\infty$ for the conditional distribution of the process $\bl(r^{-1}\sum_{i=1}^{\lfloor t/(1- F (r))\rfloor}\xi_i, t\ge0\br)$ stopped at the time when it reaches level 1 given that the latter event occurs. The limit is an increasing jump process. It is shown to be distributed as an increasing stable L\'evy process stopped at the time when it reaches level 1 conditioned on the event this level is not overshot. Some properties of this process are studied. http://arxiv.org/abs/0712.2637 --------------------------------------------------------------- 6429. LARGE DEVIATIONS ANALYSIS FOR DISTRIBUTED ALGORITHMS IN AN ERGODIC MARKOVIAN ENVIRONMENT Francis Comets (PMA) and Francois Delarue (PMA) and Ren\'e Schott (IECN, LORIA) We provide a large deviations analysis of deadlock phenomena occurring in distributed systems sharing common resources. In our model transition probabilities of resource allocation and deallocation are time and space dependent. The process is driven by an ergodic Markov chain and is reflected on the boundary of the d-dimensional cube. In the large resource limit, we prove Freidlin-Wentzell estimates, we study the asymptotic of the deadlock time and we show that the quasi-potential is a viscosity solution of a Hamilton-Jacobi equation with a Neumann boundary condition. We give a complete analysis of the colliding 2-stacks problem and show an example where the system has a stable attractor which is a limit cycle. http://arxiv.org/abs/0712.2676 --------------------------------------------------------------- 6430. SOME UNBOUNDED FUNCTIONS OF INTERMITTENT MAPS FOR WHICH THE CENTRAL LIMIT THEOREM HOLDS J. Dedecker and C. Prieur We compute some dependence coefficients for the stationary Markov chain whose transition kernel is the Perron-Frobenius operator of an expanding map $T$ of $[0, 1]$ with a neutral fixed point. We use these coefficients to prove a central limit theorem for the partial sums of $f\circ T^i$, when $f$ belongs to a large class of unbounded functions from $[0, 1]$ to ${\mathbb R}$. We also prove other limit theorems and moment inequalities. http://arxiv.org/abs/0712.2726 --------------------------------------------------------------- 6431. GRAPH LIMITS AND EXCHANGEABLE RANDOM GRAPHS Persi Diaconis and Svante Janson We develop a clear connection between deFinetti's theorem for exchangeable arrays (work of Aldous--Hoover--Kallenberg) and the emerging area of graph limits (work of Lovasz and many coauthors). Along the way, we translate the graph theory into more classical probability. http://arxiv.org/abs/0712.2749 --------------------------------------------------------------- 6432. COVERAGE PROCESSES ON SPHERES AND CONDITION NUMBERS FOR LINEAR PROGRAMMING Peter Buergisser and Felipe Cucker and Martin Lotz This paper has two agendas. Firstly, we exhibit new results for coverage processes. Let $p(n,m,\a)$ be the probability that $n$ spherical caps of angular radius~$\a$ in $S^m$ do not cover the whole sphere $S^m$. We give an exact formula for $p(n,m,\a)$ in the case $\a\in [\pi/2,\pi]$ and an upper bound for $p(n,m,\a)$ in the case $\a\in [0,\pi/2]$, which tends to $p(n,m,\pi/2)$ when $\a\to\pi/2$. In the case $\a\in [0,\pi/2]$ this yields upper bounds for the expected number of spherical caps of radius~$\a$ that are needed to cover $S^m$. Secondly, we study the condition number $\CC(A)$ of the linear programming feasibility problem $\exists x\in\R^{m+1}\, Ax\le 0,\, x\ne 0$ where $A\in\R^{n\times (m+1)}$ is randomly chosen according to the standard normal distribution. We exactly determine the distribution of $\CC(A)$ conditioned to~$A$ being feasible and provide an upper bound on the distribution function in the infeasible case. Using these results, we show that $\bE(\ln\CC (A))\le 2\ln(m+1) + 3.31$ for all $n>m$, the sharpest bound for this expectancy as of today. Both agendas are related through a result which translates between coverage and condition. http://arxiv.org/abs/0712.2816 --------------------------------------------------------------- 6433. DIRECTED PERCOLATION IN WIRELESS NETWORKS WITH INTERFERENCE AND NOISE Zhenning Kong and Edmund M. Yeh Previous studies of connectivity in wireless networks have focused on undirected geometric graphs. More sophisticated models such as Signal-to-Interference-and-Noise-Ratio (SINR) model, however, usually leads to directed graphs. In this paper, we study percolation processes in wireless networks modelled by directed SINR graphs. We first investigate interference-free networks, where we define four types of phase transitions and show that they take place at the same time. By coupling the directed SINR graph with two other undirected SINR graphs, we further obtain analytical upper and lower bounds on the critical density. Then, we show that with interference, percolation in directed SINR graphs depends not only on the density but also on the inverse system processing gain. We also provide bounds on the critical value of the inverse system processing gain. http://arxiv.org/abs/0712.2469 --------------------------------------------------------------- 6434. THE COPIES OF ANY PERMUTATION PATTERN ARE ASYMPTOTICALLY NORMAL Miklos Bona We prove that the number of copies of any given permutation pattern $q$ has an asymptotically normal distribution in random permutations. http://arxiv.org/abs/0712.2792 --------------------------------------------------------------- 6435. THE SKOROKHOD PROBLEM IN A TIME-DEPENDENT INTERVAL Krzysztof Burdzy and Weining Kang and Kavita Ramanan We consider the Skorokhod problem in a time-varying interval. We prove existence and uniqueness for the solution. We also express the solution in terms of an explicit formula. Moving boundaries may generate singularities when they touch. We establish two sets of sufficient conditions on the moving boundaries that guarantee that the variation of the local time of the associated reflected Brownian motion is, respectively, finite and infinite. We also apply these results to study the semimartingale property of a class of two-dimensional reflected Brownian motions. http://arxiv.org/abs/0712.2863 --------------------------------------------------------------- 6436. ATTRACTIVE NEAREST-NEIGHBOR SPIN SYSTEMS ON THE INTEGERS IN A RANDOMLY EVOLVING ENVIRONMENT Marcus Warfheimer We consider spin systems on the integers (i.e. interacting particle systems on the integers in which each coordinate only has two possible values and only one coordinate changes in each transition) whose rates are determined by another process, called a background process. A canonical example is the so called contact process in randomly evolving environment (CPREE), introduced and analysed by E. Broman and furthermore studied by J. Steif and the author, where the marginals of the background process independently evolve as 2- state Markov chains and determine the recovery rates for a contact process. We prove a generalization of a result by Liggett, that under certain conditions on the rates there are only two extremal stationary distributions. http://arxiv.org/abs/0712.2929 --------------------------------------------------------------- 6437. STEIN'S METHOD ON WIENER CHAOS Ivan Nourdin (PMA) and Giovanni Peccati (LSTA) We combine Malliavin calculus with Stein's method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. We also prove results concerning random variables admitting a possibly infinite Wiener chaotic decomposition. Our approach generalizes, refines and unifies the central and non-central limit theorems for multiple Wiener-It\^o integrals recently proved (in several papers, from 2005 to 2007) by Nourdin, Nualart, Ortiz- Latorre, Peccati and Tudor. We apply our techniques to prove Berry-Esseen bounds in the Breuer-Major CLT for subordinated functionals of fractional Brownian motion. By using the well-known Mehler's formula for Ornstein-Uhlenbeck semigroups, we also recover a result recently proved by Chatterjee, in the context of limit theorems for linear statistics of eigenvalues of random matrices. http://arxiv.org/abs/0712.2940 --------------------------------------------------------------- 6438. HOMOGENIZATION OF REFLECTED SEMILINEAR PDE WITH NONLINEAR NEUMANN BOUNDARY CONDITION Auguste Aman (LMAI) and Modeste Nzi We study the homogenization problem of one valued semi linear refected partial dif- ferential equation (reflected PDE for short) with nonlinear Neumann condition. The non- linear term is a function of the solution but not of its gradient. The proof are fully probabilistic and use weak convergence of an associated reflected generalized backward differential stochastic equation (reflected GBSDE in short). We also give an homogeniza- tion property for solution of semi linear reflected PDE with Neumann boundary condition in Sobolev space. http://arxiv.org/abs/0712.2986 --------------------------------------------------------------- 6439. SLE AND THE FREE FIELD: PARTITION FUNCTIONS AND COUPLINGS Julien Dubedat Schramm-Loewner Evolutions ($\SLE$) are random curves in planar simply connected domains; the massless (Euclidean) free field in such a domain is a random distribution. Both have conformal invariance properties in law. In the present article, some relations between the two objects are studied. We establish identities of partition functions between different versions of $\SLE$ and the free field with appropriate boundary conditions; this involves $\zeta$-regularization and the Polyakov-Alvarez conformal anomaly formula. We proceed with a construction of couplings of $\SLE$ with the free field, showing that, in a precise sense, chordal $\SLE$ is the solution of a stochastic "differential" equation driven by the free field. Existence and uniqueness in law for these SDEs are proved for general $\kappa>0$; pathwise uniqueness is proved for chordal $\SLE_4$. http://arxiv.org/abs/0712.3018 --------------------------------------------------------------- 6440. WHEN DO RANDOM SUBSETS DECOMPOSE A FINITE GROUP? Ariel Yadin Let A,B be two random subsets of a finite group G. We consider the event that the products of elements from A and B span the whole group; i.e. (AB union BA) = G. The study of this event gives rise to a group invariant we call \Theta(G). \Theta(G) is between 1/2 and 1, and is 1 if and only if the group is abelian. We show that a phase transition occurs as the size of A and B passes \sqrt{\Theta(G)|G|\log|G|}; i.e. for any c>0, if the size of A and B is less than (1-c)\sqrt{\Theta(G)|G|\log|G|}, then with high probability (AB union BA) does not equal G. If A and B are larger than (1+c)\sqrt{\Theta(G)|G| \log|G|} then (AB union BA) equals G with high probability. http://arxiv.org/abs/0712.3019 --------------------------------------------------------------- 6441. IDENTITIES AND INEQUALITIES FOR TREE ENTROPY Russell Lyons The notion of tree entropy was introduced by the author as a normalized limit of the number of spanning trees in finite graphs, but is defined on random infinite rooted graphs. We give some new expressions for tree entropy and use one of them to prove that tree entropy respects stochastic domination. We also prove that tree entropy is non-negative in the unweighted case. http://arxiv.org/abs/0712.3035 --------------------------------------------------------------- 6442. HARNACK INEQUALITY AND STRONG FELLER PROPERTY FOR STOCHASTIC FAST-DIFFUSION EQUATIONS Wei Liu and Feng-Yu Wang This paper presents analogous results for stochastic fast-diffusion equations. Since the fast-diffusion equation possesses weaker dissipativity than the porous medium one does, some technical difficulties appear in the study. As a compensation to the weaker dissipativity condition, a Sobolev-Nash inequality is assumed for the underlying self-adjoint operator in applications. Some concrete examples are constructed to illustrate the main results. http://arxiv.org/abs/0712.3136 --------------------------------------------------------------- 6443. TRANSPORTATION COST INEQUALITY ON PATH SPACES WITH UNIFORM DISTANCE Shizan Fang and Feng-Yu Wang and Bo Wu Starting from a sequence of independent Wright-Fisher diffusion processes on $[0,1]$, we construct a class of reversible infinite dimensional diffusion processes on $\DD_\infty:= \{{\bf x}\in Let $M$ be a complete Riemnnian manifold and $\mu$ the distribution of the diffusion process generated by $\ff 1 2\DD+Z$ where $Z$ is a $C^1$- vector field. When $\Ric-\nn Z$ is bounded below and $Z$ has, for instance, linear growth, the transportation-cost inequality with respect to the uniform distance is established for $\mu$ on the path space over $M$. A simple example is given to show the optimality of the condition. http://arxiv.org/abs/0712.3139 --------------------------------------------------------------- 6444. FROM SUPER POINCAR\'E TO WEIGHTED LOG-SOBOLEV AND ENTROPY- COST INEQUALITIES Feng-Yu Wang We derive weighted log-Sobolev inequalities from a class of super Poincar\'e inequalities. As an application, the Talagrand inequality with larger distances are obtained. In particular, on a complete connected Riemannian manifold, we prove that the $\log^\dd$-Sobolev inequality with $\dd\in (1,2)$ implies the $L^{2/(2-\dd)}$-transportation cost inequality $$W^\rr_{2/(2-\dd)}(f\mu,\mu)^{2/(2-\dd)}\le C\mu(f\log f), \mu(f) =1, f\ge 0$$ for some constant $C>0$, and they are equivalent if the curvature of the corresponding generator is bounded below. Weighted log-Sobolev and entropy-cost inequalities are also derived for a large class of probability measures on $\R^d$. http://arxiv.org/abs/0712.3142 --------------------------------------------------------------- 6445. LOG-SOBOLEV INEQAULITIES: DIFFERENT ROLES OF RIC AND HESS Feng-Yu Wang Let $P_t$ be the diffusion semigroup generated by $L:= \DD+\nn V$ on a complete connected Riemannian manifold with $\Ric\ge -(\si^2 \rr_o^2 +c)$ for some constants $\si, c>0$ and $ \rr_o$ the Riemannian distance to a fixed point. It is shown that $P_t$ is hypercontractive, or the log-Sobolev inequality holds for the associated Dirichlet form, provided $-\Hess_V\ge \dd$ holds outside of a compact set for some constant $\dd>(1+\ss 2)\si\ss{d-1}.$ This indicates, at least in finite dimensions, that $\Ric$ and $-\Hess_V$ play quite different roles for the log-Sobolev inequality to hold. The supercontractivity and the ultracontractivity are also studied. http://arxiv.org/abs/0712.3143 --------------------------------------------------------------- 6446. INTRINSIC ULTRACONTRACTIVITY ON RIEMANNIAN MANIFOLDS WITH INFINITE VOLUME MEASURES Feng-Yu Wang By establishing the intrinsic super-Poincar\'e inequality, some explicit conditions are presented for diffusion semigroups on a non-compact complete Riemannian manifold to be intrinsically ultracontractive. These conditions, as well as the resulting uniform upper bounds on the intrinsic heat kernels, are sharp for some concrete examples. http://arxiv.org/abs/0712.3144 --------------------------------------------------------------- 6447. SIMULATION OF A LOCAL TIME FRACTIONAL STABLE MOTION Matthieu Marouby In this paper, we simulate sample paths of a class of symmetric $\alpha$-stable processes using their series expression. We will develop a result in the approximation of shot-noise series. And finally, we will get a convergence rate for the approximation. http://arxiv.org/abs/0712.3210 --------------------------------------------------------------- 6448. WEAKLY DEPENDENT CHAINS WITH INFINITE MEMORY Paul Doukhan (CREST and CES) and Olivier Wintenberger (CES and SAMOS) We prove the existence of a weakly dependent strictly stationary solution of the equation $ X_t=F(X_{t-1},X_{t-2},X_{t-3},...;\xi_t)$ called {\em chain with infinite memory}. Here the {\em innovations} $\xi_t$ constitute an independent and identically distributed sequence of random variables. The function $F$ takes values in some Banach space and satisfies a Lipschitz-type condition. We also study the interplay between the existence of moments and the rate of decay of the Lipschitz coefficients of the function $F$. With the help of the weak dependence properties, we derive Strong Laws of Large Number, a Central Limit Theorem and a Strong Invariance Principle. http://arxiv.org/abs/0712.3231 --------------------------------------------------------------- 6449. SCHRAMM-LOEWNER EVOLUTION Gregory F. Lawler This is the first expository set of notes on SLE I have written since publishing a book two years ago [45]. That book covers material from a year-long class, so I cannot cover everything there. However, these notes are not just a subset of those notes, because there is a slight change of perspective. The main differences are: o I have defined SLE as a finite measure on paths that is not necessarily a probability measure. This seems more natural from the perspective of limits of lattice systems and seems to be more useful when extending SLE to non- simply connected domains. (However, I do not discuss non-simply connected domains in these notes.) o I have made more use of the Girsanov theorem in studying corresponding martingales and local martingales. As in [45], I will focus these notes on the continuous process SLE and will not prove any results about convergence of discrete processes to SLE. However, my first lecture will be about discrete processes -- it is very hard to appreciate SLE if one does not understand what it is trying to model. http://arxiv.org/abs/0712.3256 --------------------------------------------------------------- 6450. DIMENSION AND NATURAL PARAMETRIZATION FOR SLE CURVES Gregory F. Lawler Some possible definitions for the natural parametrization of SLE (Schramm-Loewner evolution) paths are proposed in terms of various limits. One of the definitions is used to give a new proof of the Hausdorff dimension of SLE paths. http://arxiv.org/abs/0712.3263 --------------------------------------------------------------- 6451. ON THE SPECTRUM OF LAMPLIGHTER GROUPS AND PERCOLATION CLUSTERS Franz Lehner and Markus Neuhauser and Wolfgang Woess Let $G$ be a finitely generated group and $X$ its Cayley graph with respect to a finite, symmetric generating set $S$. Furthermore, let $H$ be a finite group and $H \wr G$ the lamplighter group (wreath product) over $G$ with group of "lamps" $H$. We show that the spectral measure (Plancherel measure) of any symmetric "switch--walk--switch" random walk on $H \wr G$ coincides with the expected spectral measure (integrated density of states) of the random walk with absorbing boundary on the cluster of the group identity for Bernoulli site percolation on $X$ with parameter $p = 1/|H|$. The return probabilities of the lamplighter random walk coincide with the expected (annealed) return probabilites on the percolation cluster. In particular, if the clusters of percolation with parameter $p$ are almost surely finite then the spectrum of the lamplighter group is pure point. This generalizes results of Grigorchuk and Zuk, resp. Dicks and Schick regarding the case when $G$ is infinite cyclic. Analogous results relate bond percolation with another lamplighter random walk. In general, the integrated density of states of site (or bond) percolation with arbitrary parameter $p$ is always related with the Plancherel measure of a convolution operator by a signed measure on $H \wr G$, where $H = Z$ or another suitable group. http://arxiv.org/abs/0712.3135 --------------------------------------------------------------- 6452. PROBABILISTIC ANALYSIS OF THE UPWIND SCHEME FOR TRANSPORT Francois Delarue (PMA) and Fr\'ed\'eric Lagouti\`ere (LJLL) We provide a probabilistic analysis of the upwind scheme for multi-dimensional transport equations. We associate a Markov chain with the numerical scheme and then obtain a backward representation formula of Kolmogorov type for the numerical solution. We then understand that the error induced by the scheme is governed by the fluctuations of the Markov chain around the characteristics of the flow. We show, in various situations, that the fluctuations are of diffusive type. As a by-product, we prove that the scheme is of order 1/2 for an initial datum in BV and of order 1/2-a, for all a>0, for a Lipschitz continuous initial datum. Our analysis provides a new interpretation of the numerical diffusion phenomenon. http://arxiv.org/abs/0712.3217 --------------------------------------------------------------- 6453. ANALYSIS OF THE OPTIMAL EXERCISE BOUNDARY OF AMERICAN OPTIONS FOR JUMP DIFFUSIONS Erhan Bayraktar and Hao Xing In this paper we show that the optimal exercise boundary/free boundary of the American put option pricing problem for jump diffusions is continuously differentiable (except at the maturity). We also discuss its higher regularity. http://arxiv.org/abs/0712.3323 --------------------------------------------------------------- 6454. RAPID PATHS IN VON NEUMANN-GALE DYNAMICAL SYSTEMS Wael Bahsoun and Igor V. Evstigneev and Michael I. Taksar The paper examines random dynamical systems related to the classical von Neumann and Gale models of economic growth. Such systems are defined in terms of multivalued operators in spaces of random vectors, possessing certain properties of convexity and homogeneity. A central role in the theory of von Neumann-Gale dynamics is played by a special class of paths called rapid (they maximize properly defined growth rates). Up to now the theory lacked quite satisfactory results on the existence of such paths. This work provides a general existence theorem holding under assumptions analogous to the standard deterministic ones. The result solves a problem that remained open for more than three decades. http://arxiv.org/abs/0712.3353 --------------------------------------------------------------- 6455. INCORPORATING EXCHANGE RATE RISK INTO PDS AND ASSET CORRELATIONS Dirk Tasche Intuitively, the default risk of a single borrower is higher when her or his assets and debt are denominated in different currencies. Additionally, the default dependence of borrowers with assets and debt in different currencies should be stronger than in the one-currency case. By combining well- known models by Merton (1974), Garman and Kohlhagen (1983), and Vasicek (2002) we develop simple representations of PDs and asset correlations that take into account exchange rate risk. From these results, consistency conditions can be derived that link the changes in PD and asset correlation and do not require knowledge of hard-to-estimate parameters like asset value volatility. http://arxiv.org/abs/0712.3363 --------------------------------------------------------------- 6456. SCALING LIMITS FOR INTERNAL AGGREGATION MODELS WITH MULTIPLE SOURCES Lionel Levine and Yuval Peres We study the scaling limits of three different aggregation models on Z^d: internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each site distributes its excess mass equally among its neighbors. As the lattice spacing tends to zero, all three models are found to have the same scaling limit, which we describe as the solution to a certain PDE free boundary problem in R^d. In particular, internal DLA has a deterministic scaling limit. We find that the scaling limits are quadrature domains, which have arisen independently in many fields such as potential theory and fluid dynamics. Our results apply both to the case of multiple point sources and to the Diaconis-Fulton smash sum of domains. http://arxiv.org/abs/0712.3378 --------------------------------------------------------------- 6457. TAUBERIAN THEOREMS AND LARGE DEVIATIONS N. H. Bingham The link between Tauberian theorems and large deviations is surveyed, with particular reference to regular variation. http://arxiv.org/abs/0712.3410 --------------------------------------------------------------- 6458. A NOTE ON THE SUPREMUM OF A STABLE PROCESS R. A. Doney If $X$ is a spectrally positive stable process of index $\alpha\in (1,2)$ whose L\'{e}vy measure has density $cx^{-\alpha-1}$ on $(0,\infty),$ and $S_1=\sup_{0x)\backsim c\alpha^{-1}x^{-\alpha}$ as $x\to\infty.$ It is also known that $S_1 $has a continuous density, $s$ say. The point of this note is to show that $s(x)\backsim cx^{-(\alpha+1)}$ as $x\to\infty.$ http://arxiv.org/abs/0712.3414 --------------------------------------------------------------- 6459. PREDICTING THE LAST ZERO OF BROWNIAN MOTION WITH DRIFT J. du Toit and G. Peskir and A. N. Shiryaev Given a standard Brownian motion $B^{\mu}=(B_t^{\mu})_{0\le t\le T} $ with drift $\mu \in IR$ and letting $g$ denote the last zero of $B^{\mu}$ before $T$, we consider the optimal prediction problem V_*=\inf_{0\le \tau \le T}\mathsf {E}\:|\:g-\tau | where the infimum is taken over all stopping times $\tau$ of $B^ {\mu}$. Reducing the optimal prediction problem to a parabolic free-boundary problem and making use of local time-space calculus techniques, we show that the following stopping time is optimal: \tau_*=\inf {t\in [0,T] | B_t^{\mu} \le b_-(t) or B_t^{\mu} \ge b_+ (t)} where the function $t\mapsto b_-(t)$ is continuous and increasing on $[0,T]$ with $b_-(T)=0$, the function $t\mapsto b_+(t)$ is continuous and decreasing on $[0,T]$ with $b_+(T)=0$, and the pair $b_-$ and $b_+$ can be characterised as the unique solution to a coupled system of nonlinear Volterra integral equations. This also yields an explicit formula for $V_*$ in terms of $b_-$ and $b_+$. If $\mu=0$ then $b_-=-b_+$ and there is a closed form expression for $b_{\pm}$ as shown in [10] using the method of time change from [4]. The latter method cannot be extended to the case when $\mu \ne 0$ and the present paper settles the remaining cases using a different approach. http://arxiv.org/abs/0712.3415 --------------------------------------------------------------- 6460. STOCHASTIC HOMOGENIZATION OF REFLECTED DIFFUSION PROCESSES Remi Rhodes We investigate stochastic homogenization for Reflected Stochastic differential Equations on a half-plane. Our method relies on solving the "third boundary value problem" stated on a random medium and on a sector condition for the natural random Dirichlet form associated to the reflection term. http://arxiv.org/abs/0712.3416 --------------------------------------------------------------- 6461. CUMULATIVE RECORD TIMES IN A POISSON PROCESS Charles M. Goldie and Rudolf Gr\"ubel We obtain a strong law of large numbers and a functional central limit theorem, as $t\to\infty$, for the number of records up to time $t$ and the Lebesgue measure (length) of the subset of the time interval $[0,t]$ during which the Poisson process is in a record lifetime. http://arxiv.org/abs/0712.3420 --------------------------------------------------------------- 6462. LARGE DEVIATIONS FOR DIRECTED PERCOLATION ON A THIN RECTANGLE Jean-Paul Ibrahim Following the recent investigations of J. Baik and T. Suidan in \cite{baik2005gcl} and J. Martin and T. Bodineau in \cite {bodineau2005upl}, we prove large deviations properties for a last-passage percolation model in $\Z^{2}_{+}$ whose paths are close to the axis. The results are obtained for Gaussian as well as bounded weights and rely, as in \cite {baik2005gcl} and \cite{bodineau2005upl}, on a Skorokhod embedding in Brownian paths. http://arxiv.org/abs/0712.3421 --------------------------------------------------------------- 6463. LARGE-N LIMIT OF CROSSING PROBABILITIES, DISCONTINUITY, AND ASYMPTOTIC BEHAVIOR OF THRESHOLD VALUES IN MANDELBROT'S FRACTAL PERCOLATION PROCESS Erik I. Broman and Federico Camia We study Mandelbrot's percolation process in dimension $d \geq 2$. The process generates random fractal sets by an iterative procedure which starts by dividing the unit cube $[0,1]^d$ in $N^d$ subcubes, and independently retaining or discarding each subcube with probability $p$ or $1-p$ respectively. This step is then repeated within the retained subcubes at all scales. As $p$ is varied, there is a percolation phase transition in terms of paths for all $d \geq 2$, and in terms of $(d-1)$-dimensional "sheets" for all $d \geq 3$. For any $d \geq 2$, we consider the random fractal set produced at the path-percolation critical value $p_c(N,d)$, and show that the probability that it contains a path connecting two opposite faces of the cube $[0,1]^d $ tends to one as $N \to \infty$. As an immediate consequence, we obtain that the above probability has a discontinuity, as a function of $p$, at $p_c(N,d)$ for all $N$ sufficiently large. This had previously been proved only for $d=2 $ (for any $N \geq 2$). For $d \geq 3$, we prove analogous results for sheet- percolation. In dimension two, Chayes and Chayes proved that $p_c(N,2)$ converges, as $N \to \infty$, to the critical density $p_c$ of site percolation on the square lattice. Assuming the existence of the correlation length exponent $ \nu$ for site percolation on the square lattice, we establish the speed of convergence up to a logarithmic factor. In particular, our results imply that $p_c(N,2)-p_c=(\frac{1}{N})^{1/\nu+o(1)}$ as $N \to \infty$. http://arxiv.org/abs/0712.3422 --------------------------------------------------------------- 6464. AN ANALYSIS OF TWO MODIFICATIONS OF THE PETERSBURG GAME Anders Martin-L\"of Two modifications of the Petersburg game are considered: 1. Truncation, so that the player has a finite capital at his disposal. 2. A cost of borrowing capital, so that the player has to pay interest on the capital needed. In both cases limit theorems for the total net gain are derived, so that it is easy to judge if the game is favourable or not. http://arxiv.org/abs/0712.3424 --------------------------------------------------------------- 6465. A POLYMER IN A MULTI-INTERFACE MEDIUM Francesco Caravenna and Nicolas P\'etr\'elis We consider a model for a polymer chain interacting with a sequence of equi-spaced flat interfaces through a pinning potential. The intensity \delta \in R of the pinning interaction is constant, while the interface spacing T = T_N is allowed to vary with the size N of the polymer. Our main result is the explicit determination of the scaling behavior of the model in the large N limit, as a function of (T_N)_N and for fixed \delta > 0. In particular, we show that a transition occurs at T_N = O(\log N). Our approach is based on renewal theory. http://arxiv.org/abs/0712.3426 --------------------------------------------------------------- 6466. ON FINANCIAL MARKETS BASED ON TELEGRAPH PROCESSES Nikita Ratanov and Alexander Melnikov The paper develops a new class of financial market models. These models are based on generalized telegraph processes: Markov random flows with alternating velocities and jumps occurring when the velocities are switching. While such markets may admit an arbitrage opportunity, the model under consideration is arbitrage-free and complete if directions of jumps in stock prices are in a certain correspondence with their velocity and interest rate behaviour. An analog of the Black-Scholes fundamental differential equation is derived, but, in contrast with the Black-Scholes model, this equation is hyperbolic. Explicit formulas for prices of European options are obtained using perfect and quantile hedging. http://arxiv.org/abs/0712.3428 --------------------------------------------------------------- 6467. DOMAINS OF ATTRACTION OF THE RANDOM VECTOR $(X,X^2)$ AND APPLICATIONS Edward Omey Many statistics are based on functions of sample moments. Important examples are the sample variance $s_{n-1}^2$, the sample coefficient of variation SV(n), the sample dispersion SD(n) and the non-central $t$-statistic $t(n)$. The definition of these quantities makes clear that the vector defined by (\sum_{i=1}^nX_i,\sum_{i=1}^nX_i^2) plays an important role. In studying the asymptotic behaviour of this vector we start by formulating best possible conditions under which the vector $(X,X^2)$ belongs to a bivariate domain of attraction of a stable law. This approach is new, uniform and simple. Our main results include a full discussion of the asymptotic behaviour of SV(n), SD(n) and $t^2(n)$. For simplicity, in restrict ourselves to positive random variables $X$. http://arxiv.org/abs/0712.3440 --------------------------------------------------------------- 6468. MULTIVARIATE REGULAR VARIATION ON CONES: APPLICATION TO EXTREME VALUES, HIDDEN REGULAR VARIATION AND CONDITIONED LIMIT LAWS Sidney I. Resnick We attempt to bring some modest unity to three subareas of heavy tail analysis and extreme value theory: limit laws for componentwise maxima of iid random variables;hidden regular variation and asymptotic independence;conditioned limit laws when one component of a random vector is extreme. The common theme is multivariate regular variation on a cone and the three cases cited come from specifying the cones $[0,\infty]^d\setminus \{\boldsymbol 0\};(0,\infty]^d;$ and $[0,\infty]\times (0,\infty]$. http://arxiv.org/abs/0712.3442 --------------------------------------------------------------- 6469. PRISCILLA GREENWOOD: QUEEN OF PROBABILITY I.V. Evstigneev and N.H. Bingham This article contains the introduction to the special volume of Stochastics dedicated to Priscilla Greenwood, her CV and her list of publications. http://arxiv.org/abs/0712.3459 --------------------------------------------------------------- 6470. MARTINGALES AND FIRST PASSAGE TIMES OF AR(1) SEQUENCES Alexander Novikov and Nino Kordzakhia Using the martingale approach we find sufficient conditions for exponential boundedness of first passage times over a level for ergodic first order autoregressive sequences (AR(1)). Further, we prove a martingale identity to be used in obtaining explicit bounds for the expectation of first passage times. http://arxiv.org/abs/0712.3468 --------------------------------------------------------------- 6471. SMART EXPANSION AND FAST CALIBRATION FOR JUMP DIFFUSION Eric Benhamou (LJK) and Emmanuel Gobet (LJK) and Mohammed Miri (LJK) Using Malliavin calculus techniques, we derive an analytical formula for the price of European options, for any model including local volatility and jump Poisson process. We show that the accuracy of the formula depends on the smoothness of the payoff. Our approach relies on an asymptotic expansion related to small diffusion and small jump frequency. As a consequence, the calibration of such model becomes very fast. http://arxiv.org/abs/0712.3485 --------------------------------------------------------------- 6472. TRANSFORMATIONS OF L\'EVY PROCESSES Michael Sch\"urmann and Michael Skeide and Silvia Volkwardt A L\'evy process on a *-bialgebra is given by its generator, a conditionally positive hermitian linear functional vanishing at the unit element. A *-algebra homomorphism k from a *-bialgebra C to a *-bialgebra B with the property that k respects the counits maps generators on B to generators on C. A tranformation between the corrresponding two L\'evy processes is given by forming infinitesimal convolution products. This general result is applied to various situations, e.g. to a *- bialgebra and its associated primitive tensor *-bialgebra (called "generator process") as well as its associated group-like *-bialgebra (called Weyl-*- bialgebra). It follows that a L\'evy process on a *-bialgebra can be realized on Bose Fock space as the infinitesimal convolution product of its generator process such that the vacuum vector is cyclic for the L\e'vy process. Moreover, we obtain convolution approximations of the Az\'ema martingale by the Wiener process and vice versa. http://arxiv.org/abs/0712.3504 --------------------------------------------------------------- 6473. NEGATIVE CORRELATION AND LOG-CONCAVITY Jeff Kahn and Michael Neiman We settle, mostly in the negative, a number of conjectures of R. Pemantle and D. Wagner concerning negative correlation and log-concavity properties for probability measures and relations between them. Most of the negative results have also been obtained, independently but somewhat earlier, by Borcea et al. We also give short proofs of a pair of results due to Pemantle and Borcea at al.; prove that "almost exchangeable" measures satisfy the "Feder- Mihail" property, thus providing the first "non-obvious" example of a class of measures for which this important property can be shown to hold; and mention some further questions. http://arxiv.org/abs/0712.3507 --------------------------------------------------------------- 6474. ARBITRAGE FREE COINTEGRATED MODELS IN GAS AND OIL FUTURE MARKETS Gr\'egory Benmenzer (LJK) and Emmanuel Gobet (LJK) and C\'eline J \'erusalem (LJK) In this article we present a continuous time model for natural gas and crude oil future prices. Its main feature is the possibility to link both energies in the long term and in the short term. For each energy, the future returns are represented as the sum of volatility functions driven by motions. Under the risk neutral probability, the motions of both energies are correlated Brownian motions while under the historical probability, they are cointegrated by a Vectorial Error Correction Model. Our approach is equivalent to defining the market price of risk. This model is free of arbitrage: thus, it can be used for risk management as well for option pricing issues. Calibration on European market data and numerical simulations illustrate well its behavior. http://arxiv.org/abs/0712.3537 --------------------------------------------------------------- 6475. UNIVERSALITY IN TWO-DIMENSIONAL ENHANCEMENT PERCOLATION Federico Camia We consider a type of dependent percolation introduced by Aizenman and Grimmett, who showed that certain "enhancements" of independent (Bernoulli) percolation, called essential, make the percolation critical probability strictly smaller. In this paper we first prove that, for two-dimensional enhancements with a natural monotonicity property, being essential is also a necessary condition to shift the critical point. We then show that (some) critical exponents and the scaling limit of crossing probabilities of a two-dimensional percolation process are unchanged if the process is subjected to a monotonic enhancement that is not essential. This proves a form of universality for all dependent percolation models obtained via a monotonic enhancement (of Bernoulli percolation) that does not shift the critical point. For the case of site percolation on the triangular lattice, we also prove a stronger form of universality by showing that the full scaling limit is not affected by any monotonic enhancement that does not shift the critical point. http://arxiv.org/abs/0712.3412 --------------------------------------------------------------- 6476. REPEATED QUANTUM INTERACTIONS AND UNITARY RANDOM WALKS St\'ephane Attal (ICJ) and Ameur Dhahri (CEREMADE) Among the discrete evolution equations describing a quantum system $\rH_S$ undergoing repeated quantum interactions with a chain of exterior systems, we study and characterize those which are directed by classical random variables in $\RR^N$. The characterization we obtain is entirely algebraical in terms of the unitary operator driving the elementary interaction. We show that the solutions of these equations are then random walks on the group $U (\rH_0)$ of unitary operators on $\rH_0$. http://arxiv.org/abs/0712.3417 --------------------------------------------------------------- 6477. STATISTICAL PROPERTIES OF PAULI MATRICES GOING THROUGH NOISY CHANNELS St\'ephane Attal (ICJ) and Nadine Guillotin-Plantard (ICJ) We study the statistical properties of the triplet $(\sigma_x,\sigma_y,\sigma_z)$ of Pauli matrices going through a sequence of noisy channels, modeled by the repetition of a general, trace- preserving, completely positive map. We show a non-commutative central limit theorem for the distribution of this triplet, which shows up a 3-dimensional Brownian motion in the limit with a non-trivial covariance matrix. We also prove a large deviation principle associated to this convergence, with an explicit rate function depending on the stationary state of the noisy channel. http://arxiv.org/abs/0712.3418 --------------------------------------------------------------- 6478. SPECIAL, CONJUGATE AND COMPLETE SCALE FUNCTIONS FOR SPECTRALLY NEGATIVE L\'EVY PROCESSES Andreas E. Kyprianou and V\'i ctor Rivero Following from recent developments by Hubalek and Kyprianou, the objective of this paper is to provide further methods for constructing new families of scale functions for spectrally negative L\'evy processes which are completely explicit. This is the result of an observation in the aforementioned paper which permits feeding the theory of Bernstein functions directly into the Wiener-Hopf factorization for spectrally negative L\'evy processes. Many new, concrete examples of scale functions are offered although the methodology in principle delivers still more explicit examples than those listed. http://arxiv.org/abs/0712.3588 --------------------------------------------------------------- 6479. CONVERGENCE RATES FOR APPROXIMATIONS OF FUNCTIONALS OF SDES Rainer Avikainen We consider upper bounds for the approximation error E|g(X)-g(\hat X)|^p, where X and \hat X are random variables such that \hat X is an approximation of X in the L_p-norm, and the function g belongs to certain function classes, which contain e.g. functions of bounded variation. We apply the results to the approximations of a solution of a stochastic differential equation at time T by the Euler and Milstein schemes. For the Euler scheme we provide also a lower bound. http://arxiv.org/abs/0712.3635 --------------------------------------------------------------- 6480. SUBCRITICAL REGIMES IN SOME MODELS OF CONTINUUM PERCOLATION Jean-Baptiste Gou\'er\'e (MAPMO) We consider some continuum percolation models. We are mainly interested in giving some sufficient conditions for absence of percolation. We give some general conditions and then focuse on two examples. The first one is a multiscale percolation model based on the Boolean model. It was introduced by Meester and Roy and subsequently studied by Menshikov, Popov and Vachkovskaia. The second one is based on the stable marriage of Poisson and Lebesgue introduced by Hoffman, Holroyd and Peres and whose percolation properties have been studied by Freire, Popov and Vachkovskaia. This is a preliminary version: in particular, some parts of the introduction need to be developped. http://arxiv.org/abs/0712.3638 --------------------------------------------------------------- 6481. LARGE DEVIATIONS FOR EIGENVALUES OF SAMPLE COVARIANCE MATRICES Anne Fey and Remco van der Hofstad and Marten Klok We study sample covariance matrices of the form $W=\frac 1n C C^T $, where $C$ is a $k\times n$ matrix with i.i.d. mean zero entries. This is a generalization of so-called Wishart matrices, where the entries of $C$ are independent and identically distributed standard normal random variables. Such matrices arise in statistics as sample covariance matrices, and the high-dimensional case, when $k$ is large, arises in the analysis of DNA experiments. We investigate the large deviation properties of the largest and smallest eigenvalues of $W$ when either $k$ is fixed and $n\to \infty$, or $k_n \to \infty$ with $k_n=o(n/\log\log{n})$, in the case where the squares of the i.i.d. entries have finite exponential moments. Previous results, proving a.s. limits of the eigenvalues, only require finite fourth moments. Our most explicit results for $k$ large are for the case where the entries of $C$ are $\pm1$ with equal probability. We relate the large deviation rate functions of the smallest and largest eigenvalue to the rate functions for independent and identically distributed standard normal entries of $C $. This case is of particular interest, since it is related to the problem of the decoding of a signal in a code division multiple access system arising in telecommunications. In this example, $k$ plays the role of the number of users in the system, and $n$ is the length of the coding sequence of each of the users. Each user transmits at the same time and uses the same frequency, and the codes are used to distinguish the signals of the separate users. The results imply large deviation bounds for the probability of a bit error due to the interference of the various users. http://arxiv.org/abs/0712.3650 --------------------------------------------------------------- 6482. ON LIMIT THEOREMS FOR CONTINUED FRACTIONS Zbigniew S. Szewczak It is shown that for sums of functionals of digits in continued fraction expansion the Kolmogorov-Feller weak laws of large numbers and the Khinchine-L\'evy-Feller-Raikov characterization of the domain of attraction of the normal law hold. http://arxiv.org/abs/0712.3681 --------------------------------------------------------------- 6483. ON THE SPHERICITY OF SCALING LIMITS OF RANDOM PLANAR QUADRANGULATIONS Gr\'egory Marc Miermont (LM-Orsay and PMA) We give a new proof of a theorem by Le Gall & Paulin, showing that scaling limits of random planar quadrangulations are homeomorphic to the 2- sphere. The main geometric tool is a reinforcement of the notion of Gromov-Hausdorff convergence, called 1-regular convergence, that preserves topological properties of metric surfaces. http://arxiv.org/abs/0712.3687 --------------------------------------------------------------- 6484. TESSELLATIONS OF RANDOM MAPS OF ARBITRARY GENUS Gr\'egory Marc Miermont (PMA and LM-Orsay) We investigate Voronoi-like tessellations of bipartite quadrangulations on surfaces of arbitrary genus, by using a natural generalization of a bijection of Marcus and Schaeffer allowing to encode such structures into labeled maps with a fixed number of faces. We investigate the scaling limits of the latter. Applications include asymptotic enumeration results for quadrangulations, and typical metric properties of randomly sampled quadrangulations. In particular, we show that scaling limits of these random quadrangulations are such that almost every pair of points are linked by a unique geodesic. http://arxiv.org/abs/0712.3688 --------------------------------------------------------------- 6485. CENTRAL LIMIT THEOREM FOR SAMPLED SUMS OF DEPENDENT RANDOM VARIABLES Nadine Guillotin-Plantard (ICJ) and Cl\'ementine Prieur (LSProba) We prove a central limit theorem for linear triangular arrays under weak dependence conditions. Our result is then applied to the study of dependent random variables sampled by a $\bbZ$-valued transient random walk. This extends the results obtained by Guillotin-Plantard & Schneider (2003). An application to parametric estimation by random sampling is also provided. http://arxiv.org/abs/0712.3696 --------------------------------------------------------------- 6486. THE RATE OF CONVERGENCE OF SPECTRA OF SAMPLE COVARIANCE MATRICES F. G\"otze and A. Tikhomirov It is shown that the Kolmogorov distance between the spectral distribution function of a random covariance matrix $\frac1p XX^T$, where $X$ is a $n\times p$ matrix with independent entries and the distribution function of the Marchenko-Pastur law is of order $O(n^{-1/2})$. The bounds hold {\it uniformly} for any $p$, including $\frac pn$ equal or close to 1. http://arxiv.org/abs/0712.3725 --------------------------------------------------------------- 6487. PRICING AND HEDGING OF DERIVATIVES BASED ON NON-TRADABLE UNDERLYINGS Stefan Ankirchner and Peter Imkeller and Goncalo dos Reis This paper is concerned with the study of insurance related derivatives on financial markets that are based on non-tradable underlyings, but are correlated with tradable assets. We calculate exponential utility-based indifference prices, and corresponding derivative hedges. We use the fact that they can be represented in terms of solutions of forward-backward stochastic differential equations (FBSDE) with quadratic growth generators. We derive the Markov property of such FBSDE and generalize results on the differentiability relative to the initial value of their forward components. In this case the optimal hedge can be represented by the price gradient multiplied with the correlation coefficient. This way we obtain a generalization of the classical 'delta hedge' in complete markets. http://arxiv.org/abs/0712.3746 --------------------------------------------------------------- 6488. CUBATURE ON WIENER SPACE IN INFINITE DIMENSION Christian Bayer and Josef Teichmann We prove a stochastic Taylor expansion for SPDEs and apply this result to obtain cubature methods, i. e. high order weak approximation schemes for SPDEs, in the spirit of T. Lyons and N. Victoir. We can prove a high-order weak convergence for well-defined classes of test functions if the process starts at sufficiently regular points. We can also derive analogous results in the presence of L\'evy processes of finite type, here the results seem to be new even in finite dimension. Several numerical examples are added. http://arxiv.org/abs/0712.3763 --------------------------------------------------------------- 6489. LANGEVIN MOLECULAR DYNAMICS DERIVED FROM EHRENFEST DYNAMICS Anders Szepessy Stochastic Langevin molecular dynamics for nuclei is derived from quantum classical molecular dynamics, also called Ehrenfest dynamics, at positive temperature, assuming that the molecular bulk system is in equilibrium and that the initial data for the electrons is stochastically perturbed from the ground state. The initial electron probability distribution is derived from the Liouville equilibrium solution generated by the nuclei acting as a heat bath for the electrons. The diffusion and friction coefficients in the Langevin equation satisfy Einstein's fluctuation-dissipation relation. The fluctuating initial data yields, in addition to the fluctuating diffusion terms, also a contribution to the drift, modifying the standard ab initio Born- Oppenheimer solution at zero temperature, where the electrons are in their ground state for the current nuclear configuration. The dissipative friction mechanism comes from the evolution of the electron ground state, due to slow dynamics of the nuclei, while the modified drift can be understood as the mean field Born-Oppenheimer solution, for the proposed initial electron distribution at positive temperature. http://arxiv.org/abs/0712.3656 --------------------------------------------------------------- 6490. ON THE CONVERGENCE TO THE MULTIPLE WIENER-ITO INTEGRAL Xavier Bardina and Maria Jolis and Ciprian Tudor (CES and SAMOS) We study the convergence to the multiple Wiener-It\^{o} integral from processes with absolutely continuous paths. More precisely, consider a family of processes, with paths in the Cameron-Martin space, that converges weakly to a standard Brownian motion in $\mathcal C_0([0,T])$. Using these processes, we construct a family that converges weakly, in the sense of the finite dimensional distributions, to the multiple Wiener-It\^{o} integral process of a function $f\in L^2([0,T]^n)$. We prove also the weak convergence in the space $\mathcal C_0([0,T])$ to the second order integral for two important families of processes that converge to a standard Brownian motion. http://arxiv.org/abs/0712.3837 --------------------------------------------------------------- 6491. RANDOM AND INTEGRABLE MODELS IN MATHEMATICS AND PHYSICS Pierre van Moerbeke This set of Montreal lectures is an elementary and sketchy introduction to the general field of random matrices. The first half is devoted to combinatorial models, whereas the second half deals with random matrix questions(GUE, etc...). http://arxiv.org/abs/0712.3847 --------------------------------------------------------------- 6492. STOCHASTIC INTEGRATION BASED ON SIMPLE, SYMMETRIC RANDOM WALKS Tam\'as Szabados (Budapest University of Technology and Economics), Bal\'azs Sz\'ekely (Budapest University of Technology and Economics) A new approach to stochastic integration is described, which is based on an a.s. pathwise approximation of the integrator by simple, symmetric random walks. Hopefully, this method is didactically more advantageous, more transparent, and technically less demanding than other existing ones. In a large part of the theory one has a.s. uniform convergence on compacts. In particular, it gives a.s. convergence for the stochastic integral of a finite variation function of the integrator, which is not c\`adl\`ag in general. http://arxiv.org/abs/0712.3908 --------------------------------------------------------------- 6493. NOISY HETEROCLINIC NETWORKS Yuri Bakhtin We consider a white noise perturbation of dynamics in the neighborhood of a heteroclinic network. We show that under the logarithmic time rescaling the diffusion converges in distributon in a special topology to a piecewise constant process that jumps between saddle points along the heteroclinic orbits of the network. We also obtain precise asymptotics for the exit measure for a domain containing the starting point of the diffusion. http://arxiv.org/abs/0712.3952 --------------------------------------------------------------- 6494. UNIQUENESS FOR THE MARTINGALE PROBLEM ASSOCIATED WITH PURE JUMP PROCESSES OF VARIABLE ORDER Huili Tang Let $L$ be the operator defined on $C^2$ functions by $$L f(x)=\int[f(x+h)-f(x)-1_{(|h|\leq 1)}\nabla f(x)\cdot h]\frac{n(x,h)}{|h|^{d+\alpha(x)}}dh.$$ This is an operator of variable order and the corresponding process is of pure jump type. We consider the martingale problem associated with $L$. Sufficient conditions for existence and uniqueness are given. Transition density estimates for $\alpha$-stable processes are also obtained. http://arxiv.org/abs/0712.4137 --------------------------------------------------------------- 6495. EDGEWORTH EXPANSIONS IN OPERATOR FORM Zbigniew S. Szewczak An operator form of asymptotic expansions for Markov chains is established. Coefficients are given explicitly. Such expansions require a certain modification of the classical spectral method. They prove to be extremely useful within the context of large deviations. http://arxiv.org/abs/0712.4199 --------------------------------------------------------------- 6496. MARTINGALE PROOFS OF MANY-SERVER HEAVY-TRAFFIC LIMITS FOR MARKOVIAN QUEUES Guodong Pang and Rishi Talreja and Ward Whitt This is an expository review paper illustrating the ``martingale method'' for proving many-server heavy-traffic stochastic-process limits for queueing models, supporting diffusion-process approximations. Careful treatment is given to an elementary model -- the classical infinite-server model $M/M/ \infty$, but models with finitely many servers and customer abandonment are also treated. The Markovian stochastic process representing the number of customers in the system is constructed in terms of rate-1 Poisson processes in two ways: (i) through random time changes and (ii) through random thinnings. Associated martingale representations are obtained for these constructions by applying, respectively: (i) optional stopping theorems where the random time changes are the stopping times and (ii) the integration theorem associated with random thinning of a counting process. Convergence to the diffusion process limit for the appropriate sequence of scaled queueing processes is obtained by applying the continuous mapping theorem. A key FCLT and a key FWLLN in this framework are established both with and without applying martingales. http://arxiv.org/abs/0712.4211 --------------------------------------------------------------- 6497. EXCURSION SETS OF STABLE RANDOM FIELDS Robert J. Adler and Gennady Samorodnitsky and Jonathan E. Taylor Studying the geometry generated by Gaussian and Gaussian- related random fields via their excursion sets is now a well developed and well understood subject. The purely non-Gaussian scenario has, however, not been studied at all. In this paper we look at three classes of stable random fields, and obtain asymptotic formulae for the mean values of various geometric characteristics of their excursion sets over high levels. While the formulae are asymptotic, they contain enough information to show that not only do stable random fields exhibit geometric behaviour very different from that of Gaussian fields, but they also differ significantly among themselves. http://arxiv.org/abs/0712.4276 --------------------------------------------------------------- 6498. CONVOLUTION TYPE STOCHASTIC VOLTERRA EQUATIONS Anna Karczewska The aim of this work is to present, in self-contained form, results concerning fundamental and the most important questions related to linear stochastic Volterra equations of convolution type. The paper is devoted to study the existence and some kind of regularity of solutions to stochastic Volterra equations in Hilbert space and the space of tempered distributions, as well. In recent years the theory of Volterra equations, particularly fractional ones, has undergone a big development. This is an emerging area of research with interesting mathematical questions and various important applications. The increasing interest in these equations comes from their applications to problems from physics and engeenering, particularly from viscoelasticity, heat conduction in materials with memory or electrodynamics with memory. http://arxiv.org/abs/0712.4357 --------------------------------------------------------------- 6499. LIMIT THEOREMS FOR INTERNAL AGGREGATION MODELS Lionel Levine We study the scaling limits of three different aggregation models on the integer lattice Z^d: internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each site distributes its excess mass equally among its neighbors. As the lattice spacing tends to zero, all three models are found to have the same scaling limit, which we describe as the solution to a certain PDE free boundary problem in R^d. In particular, internal DLA has a deterministic scaling limit. We find that the scaling limits are quadrature domains, which have arisen independently in many fields such as potential theory and fluid dynamics. Our results apply both to the case of multiple point sources and to the Diaconis-Fulton smash sum of domains. In the special case when all particles start at a single site, we show that the scaling limit is a Euclidean ball in R^d, and give quantitative bounds on the rate of convergence to a ball. We also improve on the previously best known bounds of Le Borgne and Rossin in Z^2 and Fey and Redig in higher dimensions for the shape of the classical abelian sandpile model. Lastly, we study the sandpile group of a regular tree whose leaves are collapsed to a single sink vertex, and determine the decomposition of the full sandpile group as a product of cyclic groups. For the regular ternary tree of height n, for example, the sandpile group is isomorphic to (Z_3)^{2^{n-3}} x (Z_7)^{2^{n-4}} x ... x Z_{2^{n-1}-1} x Z_{2^n-1}. We use this result to prove that rotor-router aggregation on the regular tree yields a perfect ball. http://arxiv.org/abs/0712.4358 --------------------------------------------------------------- 6500. JUDGMENT Ruadhan O'Flanagan The concept of a judgment as a logical action which introduces new information into a deductive system is examined. This leads to a way of mathematically representing implication which is distinct from the familiar material implication, according to which "If A then B" is considered to be equivalent to "B or not-A". This leads, in turn, to a resolution of the paradox of the raven. http://arxiv.org/abs/0712.4402 --------------------------------------------------------------- 6501. THE MAXIMAL PROBABILITY THAT K-WISE INDEPENDENT BITS ARE ALL 1 Ron Peled and Ariel Yadin and Amir Yehudayoff A k-wise independent distribution on n bits is a joint distribution of the bits such that each k of them are independent. In this paper we consider k-wise independent distributions with identical marginals, each bit has probability p to be 1. We address the following question: how high can the probability that all the bits are 1 be, for such a distribution? For a wide range of the parameters n,k and p we find an explicit lower bound for this probability which matches an upper bound given by Benjamini et al., up to multiplicative factors of lower order. The question we investigate can be seen as a relaxation of a major open problem in error-correcting codes theory, namely, how large can a linear error correcting code with given parameters be? The question is a type of discrete moment problem, and our approach is based on showing that bounds obtained from the theory of the classical moment problem provide good approximations for it. The main tool we use is a bound controlling the change in the expectation of a polynomial after small perturbation of its zeros. http://arxiv.org/abs/0801.0059 --------------------------------------------------------------- 6502. FROM POWER LAWS TO FRACTIONAL DIFFUSION: THE DIRECT WAY Rudolf Gorenflo and Entsar A.A. Abdel-Rehim Starting from the model of continuous time random walk, we focus our interest on random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. By stating the relevant lemmata (of Tauber type) for the distribution functions we need not distinguish between continuous and discrete space and time. We will see that, by a well-scaled passage to the diffusion limit, generalized diffusion processes, fractional in time as well as in space, are obtained. The corresponding equation of evolution is a linear partial pseudo-differential equation with fractional derivatives in time and in space, the orders being equal to the above exponents. Such processes are well approximated and visualized by simulation via various types of random walks. For their explicit solutions there are available integral representations that allow to investigate their detailed structure. http://arxiv.org/abs/0801.0142 --------------------------------------------------------------- 6503. SOME RECENT ADVANCES IN THEORY AND SIMULATION OF FRACTIONAL DIFFUSION PROCESSES Rudolf Gorenflo and Francesco Mainardi To offer a view into the rapidly developing theory of fractional diffusion processes we describe in some detail three topics of present interest: (i) the well-scaled passage to the limit from continuous time random walk under power law assumptions to space-time fractional diffusion, (ii) the asymptotic universality of the Mittag-Leffler waiting time law in time-fractional processes, (iii) our method of parametric subordination for generating particle trajectories. http://arxiv.org/abs/0801.0146 --------------------------------------------------------------- 6504. RESOLVENT OF LARGE RANDOM GRAPHS Charles Bordenave and Marc Lelarge We analyze the convergence of the spectrum of large random graphs to the spectrum of a limit infinite graph. We apply these results to graphs converging locally to trees and derive a new formula for the Stieljes transform of the spectral measure of such graphs. We illustrate our results on the uniform regular graphs, Erdos-Renyi graphs and preferential attachment graphs. We sketch examples of application for weighted graphs, bipartite graphs and the uniform spanning tree of n vertices. http://arxiv.org/abs/0801.0155 --------------------------------------------------------------- 6505. STANDARD REPRESENTATION OF MULTIVARIATE FUNCTIONS ON A GENERAL PROBABILITY SPACE Svante Janson It is well-known that a random variable, i.e., a function defined on a probability space, with values in a Borel space, can be represented on the special probability space consisting of the unit interval with Lebesgue measure. We show an extension of this to multivariate functions. This is motivated by some recent constructions of random graphs. http://arxiv.org/abs/0801.0196 --------------------------------------------------------------- 6506. PROPERTIES OF EXPECTATIONS OF FUNCTIONS OF MARTINGALE DIFFUSIONS George Lowther Given a real valued and time-inhomogeneous martingale diffusion X, we investigate the properties of functions defined by the conditional expectation f(t,X_t)=E[g(X_T)|F_t]. We show that whenever g is monotonic or Lipschitz continuous then f(t,x) will also be monotonic or Lipschitz continuous in x. If g is convex then f(t,x) will be convex in x and decreasing in t. We also define the marginal support of a process and show that it almost surely contains the paths of the process. Although f need not be jointly continuous, we show that it will be continuous on the marginal support of X. We prove these results for a generalization of diffusion processes that we call `almost-continuous diffusions', and includes all continuous and strong Markov processes. http://arxiv.org/abs/0801.0330 --------------------------------------------------------------- 6507. OLD AND NEW EXAMPLES OF SCALE FUNCTIONS FOR SPECTRALLY NEGATIVE LEVY PROCESSES F. Hubalek and A.E. Kyprianou We give a review of the state of the art with regard to the theory of scale functions for spectrally negative Levy processes. From this we introduce a general method for generating new families of scale functions. Using this method we introduce a new family of scale functions belonging to the Gaussian Tempered Stable Convolution (GTSC) class. We give particular emphasis to special cases as well as cross-referencing their analytical behaviour against known general considerations. http://arxiv.org/abs/0801.0393 --------------------------------------------------------------- 6508. AN EFFECTIVE BOREL-CANTELLI LEMMA. CONSTRUCTING ORBITS WITH REQUIRED STATISTICAL PROPERTIES Stefano Galatolo and Mathieu Hoyrup and Cristobal Rojas In the general context of computable metric spaces and computable measures we prove a kind of constructive Borel-Cantelli lemma: given a sequence (recursive in some way) of sets $A_{i}$ with recursively summable measures, there are computable points which are not contained in infinitely many $ A_{i} $. As a consequence of this we obtain the existence of computable points which follow the typical statistical behavior of a dynamical system (they satisfy the Birkhoff theorem) for a large class of systems, having computable invariant measure and polynomial decay of correlation. This is applied to uniformly hyperbolic systems, piecewise expanding maps, systems on the interval with an indifferent fixed point and it directly implies the existence of computable numbers which are normal with respect to any base. http://arxiv.org/abs/0711.1478 --------------------------------------------------------------- 6509. DETERMINANTAL IDENTITY FOR MULTILEVEL SYSTEMS AND FINITE DETERMINANTAL PROCESSES J. Harnad and A. Yu. Orlov We give a simple algebraic derivation of a useful determinantal identity for multilevel systems such as random matrix chains and finite determinantal point processes, with applications to the calculation of point correlators, gap probabililties and Janossy densities. http://arxiv.org/abs/0712.3892 --------------------------------------------------------------- 6510. ALGORITHMICALLY RANDOM POINTS IN MEASURE PRESERVING SYSTEMS, STATISTICAL BEHAVIOUR, COMPLEXITY AND ENTROPY Stefano Galatolo and Mathieu Hoyrup and Cristobal Rojas We consider the dynamical behavior of Martin-L\"of random points in dynamical systems over metric spaces with a computable dynamics and a computable invariant measure. We use computable partitions to define a sort of effective symbolic model for the dynamics. Trough this construction we prove that such points have typical statistical behavior (the behavior which is typical in the Birkhoff ergodic theorem) and are recurrent. We introduce and compare some notion of complexity for orbits in dynamical systems and we prove that the complexity of the orbits of random points equals the entropy of the system. http://arxiv.org/abs/0801.0209 From pas at lists.imstat.org Tue Mar 4 05:05:32 2008 From: pas at lists.imstat.org (Probability Abstract Service) Date: Tue, 04 Mar 2008 12:05:32 +0100 Subject: [PAS] Probability Abstracts 102 Message-ID: <18A51445-4E14-40F3-8BFB-C4D68D2C13FA@unimi.it> Probability Abstracts 102 This document contains abstracts 6511-6752 from January-1-2008 to February-29-2008. They have been mailed on March 4th, 2008. This letter can be also found on line at http://pas.imstat.org/Letters/letter_102.shtml --------------------------------------------------------------- 6511. SOME EXAMPLES OF ABSOLUTE CONTINUITY OF MEASURES IN STOCHASTIC FLUID DYNAMICS B. Ferrario A non linear Ito equation in a Hilbert space is studied by means of Girsanov theorem. We consider a non linearity of polynomial growth in suitable norms, including that of quadratic type which appears in the Kuramoto- Sivashinsky equation and in the Navier-Stokes equation. We prove that Girsanov theorem holds for the 1-dimensional stochastic Kuramoto-Sivashinsky equation and for a modification of the 2- and 3-dimensional stochastic Navier-Stokes equation. In this way, we prove existence and uniqueness of solutions for these stochastic equations. Moreover, the asymptotic behaviour for large time is characterized. http://arxiv.org/abs/0801.0496 --------------------------------------------------------------- 6512. MAXIMUM AND ENTROPIC REPULSION FOR A GAUSSIAN MEMBRANE MODEL IN THE CRITICAL DIMENSION Noemi Kurt We consider the real-valued centered Gaussian field on the four- dimensional integer lattice, whose covariance matrix is given by the Green's function of the discrete Bilaplacian. This is interpreted as a model for a semiflexible membrane. $d=4$ is the critical dimension for this model. We discuss the effect of a hard wall on the membrane, via a multiscale analysis of the maximum of the field. We use analytic and probabilistic tools to describe the correlation structure of the field. http://arxiv.org/abs/0801.0551 --------------------------------------------------------------- 6513. RANDOM TURN WALK ON A HALF LINE WITH CREATION OF PARTICLES AT THE ORIGIN J.W. van de Leur and A. Yu. Orlov We consider a version of random motion of hard core particles on the semi-lattice $ 1, 2, 3,...$, where in each time instant one of three possible events occurs, viz., (a) a randomly chosen particle hops to a free neighboring site, (b) a particle is created at the origin (namely, at site 1) provided that site 1 is free and (c) a particle is eliminated at the origin (provided that the site 1 is occupied). Relations to the BKP equation are explained. Namely, the tau functions of two different BKP hierarchies provide generating functions respectively (I) for transition weights between different particle configurations and (II) for an important object: a normalization function which plays the role of the statistical sum for our non-equilibrium system. As an example we study a model where the hopping rate depends on two parameters ($r$ and $\beta$). For time $\time\to\infty$ we obtain the asymptotic configuration of particles obtained from the initial empty state (the state without particles) and find an analog of the first order transition at $ \beta=1$. http://arxiv.org/abs/0801.0066 --------------------------------------------------------------- 6514. SOME REMARKS ON TANGENT MARTINGALE DIFFERENCE SEQUENCES IN $L^1$- SPACES Sonja Cox and Mark Veraar Let X be a Banach space. Suppose that for all $p\in (1, \infty)$ a constant $C_{p,X}$ depending only on X and p exists such that for any two X- valued martingales f and g with tangent martingale difference sequences one has \[\E\|f\|^p \leq C_{p,X} \E\|g\|^p (*).\] This property is equivalent to the UMD condition. In fact, it is still equivalent to the UMD condition if in addition one demands that either f or g satisfy the so-called (CI) condition. However, for some applications it suffices to assume that (*) holds whenever g satisfies the (CI) condition. We show that the class of Banach spaces for which (*) holds whenever only g satisfies the (CI) condition is more general than the class of UMD spaces, in particular it includes the space L^1. We state several problems related to (*) and other decoupling inequalities. http://arxiv.org/abs/0801.0695 --------------------------------------------------------------- 6515. ARBITRAGE FREE MODELS IN MARKETS WITH TRANSACTION COSTS Erhan Bayraktar In \cite{Gua} the notion of stickiness for stochastic processes was introduced. It was also shown that stickiness implies absense of arbitrage in a market with proportional transaction costs. In this paper, we investigate the notion of stickiness further. In particular, we show that stickiness is invariant under composition with continuous functions. We also prove a time change result on stickiness. As an application we provide sufficient conditions for continuous semimartingales to be sticky (A counter example show that not all semi-martingales are sticky). As a result, our paper provides an extended class of stochastic processes that are consistent with the no arbitrage property in a market with friction. http://arxiv.org/abs/0801.0718 --------------------------------------------------------------- 6516. CONVERGENCE OF MULTI-DIMENSIONAL QUANTIZED $SDE$'S Gilles Pag\`es (PMA) and Afef Sellami (PMA) We quantize a multidimensional $SDE$ (in the Stratanovich sense) by solving the related $ODE$'s in which the Brownian motion has been replaced by the components of stationary quantizers. We make a connection with rough path theory to show that such quantizations converge toward the solution of the $SDE$. In some particular cases, we show that this procedure provide some rate optimal quantizations of the equation. http://arxiv.org/abs/0801.0726 --------------------------------------------------------------- 6517. LOWER LARGE DEVIATIONS FOR MAXIMAL FLOWS THROUGH A BOX IN FIRST PASSAGE PERCOLATION Rapha\"el Rossignol and Marie Th\'eret We consider the standard first passage percolation model in $ \mathbb{Z}^d$ for $d\geq 2$. We are interested in two quantities, the maximal flow $ \tau$ between the lower half and the upper half of the box, and the maximal flow $\phi$ between the top and the bottom of the box. A standard subadditive argument yields the law of large numbers for $\tau$. Kesten and Zhang have proved the law of large numbers for $\phi$. The two variables grow linearly with the surface $s$ of the basis of the box, with the same deterministic speed. We study the probabilities that the rescaled variables $\tau /s $ and $\phi /s$ are abnormally small. Using a concentration inequality, we show that these probabilities decay exponentially fast with $s$, when $s$ grows to infinity. Moreover, we prove an associated large deviation principle of speed $s$ for $\tau /s$, and for $\phi /s$. For $\phi$, we require either that the box is sufficiently flat, or that its sides are parallel to the coordinates hyperplanes. http://arxiv.org/abs/0801.0967 --------------------------------------------------------------- 6518. LAWS OF LARGE NUMBERS FOR CONTINUOUS BELIEF MEASURES ON COMPACT SPACES Yann Rebille We prove for outer continuous belief measures defined on compact spaces strong and weak laws of large numbers as Kolmogorov's one for measures. These results contribute to M. Marinacci's (Journal of Economic Theory 84 (1999) 145-195) though with different methods. http://arxiv.org/abs/0801.0976 --------------------------------------------------------------- 6519. IMPRECISE MARKOV CHAINS AND THEIR LIMIT BEHAVIOUR Gert de Cooman and Filip Hermans and Erik Quaeghebeur When the parameters of a finite Markov chain in discrete time, i.e., its initial and transition probabilities, are not well known, we can and should perform a sensitivity analysis. This is done by considering as basic uncertainty models the so-called credal sets that these probabilities are known or believed to belong to, and by allowing the probabilities to vary over such sets. This leads to the definition of an imprecise Markov chain. We show that the time evolution of such a system can be studied efficiently using so-called lower and upper expectations, which are equivalent mathematical representations of credal sets. We also study how the inferred credal set about the state at time n evolves as n goes to infinity, and we show that under quite unrestrictive conditions, this credal set converges to a uniquely invariant credal set, regardless of the credal set given for the initial state of the system. We thus effectively prove a Perron-Frobenius Theorem for a special class of non-linear dynamical systems in discrete time. http://arxiv.org/abs/0801.0980 --------------------------------------------------------------- 6520. LAW OF LARGE NUMBERS FOR NON-ADDITIVE MEASURES Yann Rebille Our aim is to give for some classes non-additive measures some limit theorems. For balanced games we obtain a weak and strong law of large numbers for bounded random variables, a sharper conclusion is obtain with exact games. We provide an extension to upper enveloppe measures. http://arxiv.org/abs/0801.0984 --------------------------------------------------------------- 6521. MODERATE DEVIATIONS FOR RANDOM FIELDS AND RANDOM COMPLEX ZEROES Boris Tsirelson Moderate deviations for random complex zeroes are deduced from a new theorem on moderate deviations for random fields. http://arxiv.org/abs/0801.1050 --------------------------------------------------------------- 6522. ON THE ROBUSTNESS OF POWER-LAW RANDOM GRAPHS IN THE FINITE MEAN, INFINITE VARIANCE REGION I. Norros and H. Reittu We consider a conditionally Poissonian random graph model where the mean degrees, `capacities', follow a power-tailed distribution with finite mean and infinite variance. Such a graph of size $N$ has a giant component which is super-small in the sense that the typical distance between vertices is of the order of $\log\log N$. The shortest paths travel through a core consisting of nodes with high mean degrees. In this paper we derive upper bounds of the typical distance when an upper part of the core is removed, including the case that the whole core is removed. http://arxiv.org/abs/0801.1079 --------------------------------------------------------------- 6523. FRACTIONAL BROWNIAN MOTION IN PRESENCE OF TWO FIXED ADSORBING BOUNDARIES G. Oshanin We study the long-time asymptotics of the probability P_t that the Riemann-Liouville fractional Brownian motion with Hurst index H does not escape from a fixed interval [-L,L] up to time t. We show that for any H \in ]0,1], for both subdiffusion and superdiffusion regimes, this probability obeys \ln(P_t) \sim - t^{2 H}/L^2, i.e. may decay slower than exponential (subdiffusion) or faster than exponential (superdiffusion). This implies that survival probability S_t of particles undergoing fractional Brownian motion in a one-dimensional system with randomly placed traps follows \ln(S_t) \sim - n^{2/3} t^{2H/3} as t \to \infty, where n is the mean density of traps. http://arxiv.org/abs/0801.0676 --------------------------------------------------------------- 6524. ON THE EXTREMAL RAYS OF THE CONE OF POSITIVE, POSITIVE DEFINITE FUNCTIONS Philippe Jaming (MAPMO) and Mat\'e Matolcsi and Szilard Gy. R\'evesz The aim of this paper is to investigate the cone of non-negative, radial, positive-definite functions in the set of continuous functions on $\R^d $. Elements of this cone admit a Choquet integral representation in terms of the extremals. The main feature of this article is to characterize some large classes of such extremals. In particular, we show that there many other extremals than the gaussians, thus disproving a conjecture of G. Choquet and that no reasonable conjecture can be made on the full set of extremals. The last feature of this article is to show that many characterizations of positive definite functions available in the literature are actually particular cases of the Choquet integral representations we obtain. http://arxiv.org/abs/0801.0941 --------------------------------------------------------------- 6525. IMPRECISE PROBABILITY TREES: BRIDGING TWO THEORIES OF IMPRECISE PROBABILITY Gert de Cooman and Filip Hermans We give an overview of two approaches to probability theory where lower and upper probabilities, rather than probabilities, are used: Walley's behavioural theory of imprecise probabilities, and Shafer and Vovk's game- theoretic account of probability. We show that the two theories are more closely related than would be suspected at first sight, and we establish a correspondence between them that (i) has an interesting interpretation, and (ii) allows us to freely import results from one theory into the other. Our approach leads to an account of probability trees and random processes in the framework of Walley's theory. We indicate how our results can be used to reduce the computational complexity of dealing with imprecision in probability trees, and we prove an interesting and quite general version of the weak law of large numbers. http://arxiv.org/abs/0801.1196 --------------------------------------------------------------- 6526. STOCHASTIC PROCESSES AND THEIR SPECTRAL REPRESENTATIONS OVER NON-ARCHIMEDEAN FIELDS S.V. Ludkovsky The article is devoted to stochastic processes with values in finite- and infinite-dimensional vector spaces over infinite fields $\bf K$ of zero characteristics with non-trivial non-archimedean norms. For different types of stochastic processes controlled by measures with values in $\bf K$ and in complete topological vector spaces over $\bf K$ stochastic integrals are investigated. Vector valued measures and integrals in spaces over $\bf K$ are studied. Theorems about spectral decompositions of non-archimedean stochastic processes are proved. http://arxiv.org/abs/0801.1209 --------------------------------------------------------------- 6527. OPTIMAL CO-ADAPTED COUPLING FOR THE SYMMETRIC RANDOM WALK ON THE HYPERCUBE Stephen B. Connor and Saul D. Jacka Let X and Y be two simple symmetric continuous-time random walks on the vertices of the n-dimensional hypercube. We consider the class of co- adapted couplings of these processes, and describe an intuitive coupling which is shown to be the fastest in this class. http://arxiv.org/abs/0801.1220 --------------------------------------------------------------- 6528. ON THE SINGULARITY OF RANDOM MATRICES WITH INDEPENDENT ENTRIES Laurent Bruneau and Francois Germinet We consider n by n real matrices whose entries are non-degenerate random variables that are independent but non necessarily identically distributed, and show that the probability that such a matrix is singular is O(1/ sqrt{n}). The purpose of this note is to provide a short and elementary proof of this fact using a Bernoulli decomposition of arbitrary non degenerate random variables. http://arxiv.org/abs/0801.1221 --------------------------------------------------------------- 6529. BALANCED ROUTING OF RANDOM CALLS Malwina J. Luczak and Colin McDiarmid We consider an online routing problem in continuous time, where calls have Poisson arrivals and exponential durations. The first-fit dynamic alternative routing algorithm sequentially selects up to $d$ random two-link routes between the two endpoints of a call, via an intermediate node, and assigns the call to the first route with spare capacity on each link, if there is such a route. The balanced dynamic alternative routing algorithm simultaneously selects $d$ random two-link routes; and the call is accepted on a route minimising the maximum of the loads on its two links, provided neither of these two links is saturated. We determine the capacities needed for these algorithms to route calls successfully, and find that the balanced algorithm requires a much smaller capacity. http://arxiv.org/abs/0801.1260 --------------------------------------------------------------- 6530. FINITELY ADDITIVE SUPERMARTINGALES Gianluca Cassese The concept of finitely additive supermartingales, originally due to Bochner, is revived and developed. We exploit it to study measure decompositions over filtered probability spaces and the properties of the associated Dol\'{e}ans-Dade measure. We obtain versions of the Doob Meyer decomposition and, as an application, we establish a version of the Bichteler and Dellacherie theorem with no exogenous probability measure. http://arxiv.org/abs/0801.1262 --------------------------------------------------------------- 6531. EXCHANGEABLE LOWER PREVISIONS Gert de Cooman and Erik Quaeghebeur and Enrique Miranda We extend de Finetti's (1937) notion of exchangeability to finite and countable sequences of variables, when a subject's beliefs about them are modelled using coherent lower previsions rather than (linear) previsions. We prove representation theorems in both the finite and the countable case, in terms of sampling without and with replacement, respectively. We also establish a convergence result for sample means of exchangeable sequences. Finally, we study and solve the problem of exchangeable natural extension: how to find the most conservative (point-wise smallest) coherent and exchangeable lower prevision that dominates a given lower prevision. http://arxiv.org/abs/0801.1265 --------------------------------------------------------------- 6532. GAME-THEORETIC BROWNIAN MOTION Vladimir Vovk This paper suggests a perfect-information game, along the lines of Levy's characterization of Brownian motion, that formalizes the process of Brownian motion in game-theoretic probability. This is perhaps the simplest situation where probability emerges in a non-stochastic environment. http://arxiv.org/abs/0801.1309 --------------------------------------------------------------- 6533. ON MAXIMA OF PERIODOGRAMS OF STATIONARY PROCESSES Zhengyan Lin and Weidong Liu We consider the limit distribution of maxima of periodograms for stationary processes. Our method is based on $m$-dependent approximation for stationary processes and a moderate deviation result. http://arxiv.org/abs/0801.1357 --------------------------------------------------------------- 6534. CENTRAL AND $L^2$-CONCENTRATION OF 1-LIPSCHITZ MAPS INTO $ \MATHBB{R}$-TREES Kei Funano In this paper, we examine the L\'{e}vy-Milman concentration phenomenon of 1-Lipschitz maps from mm-spaces to $\mathbb{R}$-trees. Our main theorems assert that the concentration to $\mathbb{R}$-trees follows from the concentration to the real line. http://arxiv.org/abs/0801.1371 --------------------------------------------------------------- 6535. LARGE DEVIATIONS FOR STOCHASTIC EVOLUTION EQUATIONS WITH SMALL MULTIPLICATIVE NOISE Wei Liu The Freidlin-Wentzell large deviation principle is established for the distributions of stochastic evolution equations with general monotone drift and small multiplicative noise. Roughly speaking, the assumptions one need for large deviation principle are classical monotone condition on drift part (as for the existence and uniqueness of solution) and Lipschitz condition on diffusion coefficient. As applications we can apply the main result to different type examples of SPDEs (e.g. stochastic reaction-diffusion equation, stochastic porous media and fast diffusion equations, stochastic p- Laplacian equation) in Hilbert space. The weak convergence approach is employed to verify the Laplace principle, which is equivalent to large deviation principle in our framework. http://arxiv.org/abs/0801.1443 --------------------------------------------------------------- 6536. LONGEST INCREASING SUBSEQUENCES, PLANCHEREL-TYPE MEASURE AND THE HECKE INSERTION ALGORITHM Hugh Thomas and Alexander Yong We define and study the Plancherel-Hecke probability measure on Young diagrams; the Hecke algorithm of [Buch-Kresch-Shimozono-Tamvakis-Yong '06] is interpreted as a polynomial-time exact sampling algorithm for this measure. Using the results of [Thomas-Yong '07] on jeu de taquin for increasing tableaux, a symmetry property of the Hecke algorithm is proved, in terms of longest strictly increasing/decreasing subsequences of words. This parallels classical theorems of [Schensted '61] and of [Knuth '70], respectively, on the Schensted and Robinson-Schensted-Knuth algorithms. We investigate, and conjecture about, the limit typical shape of the measure, in analogy with work of [Vershik-Kerov '77], [Logan-Shepp '77] and others on the ``longest increasing subsequence problem'' for permutations. We also include a related extension of [Aldous-Diaconis '99] on patience sorting. Together, these results provide a new rationale for the study of increasing tableau combinatorics, distinct from the original algebraic-geometric ones concerning K- theoretic Schubert calculus. http://arxiv.org/abs/0801.1319 --------------------------------------------------------------- 6537. ESTIMATION OF ORDINAL PATTERN PROBABILITIES IN FRACTIONAL BROWNIAN MOTION Mathieu Sinn and Karsten Keller For equidistant discretizations of fractional Brownian motion (fBm), the probabilities of ordinal patterns of order d=2 are monotonically related to the Hurst parameter H. By plugging the sample relative frequency of those patterns indicating changes between up and down into the monotonic relation to H, one obtains the Zero Crossing (ZC) estimator of the Hurst parameter which has found considerable attention in mathematical and applied research. In this paper, we generally discuss the estimation of ordinal pattern probabilities in fBm. As it turns out, according to the sufficiency principle, for ordinal patterns of order d=2 any reasonable estimator is an affine functional of the sample relative frequency of changes. We establish strong consistency of the estimators and show them to be asymptotically normal for H<3/4. Further, we derive confidence intervals for the Hurst parameter. Simulation studies show that the ZC estimator has larger variance but less bias than the HEAF estimator of the Hurst parameter. http://arxiv.org/abs/0801.1598 --------------------------------------------------------------- 6538. RANDOM SUBGRAPHS OF THE 2D HAMMING GRAPH: THE SUPERCRITICAL PHASE Remco van der Hofstad and Malwina J. Luczak We study random subgraphs of the 2-dimensional Hamming graph H(2,n), which is the Cartesian product of two complete graphs on $n$ vertices. Let $p$ be the edge probability, and write $p=\frac{1+\vep}{2(n-1)}$ for some $\vep \in \R$. In Borgs et al., Random subgraphs of finite graphs: I. The scaling window under the triangle condition, Rand. Struct. Alg. (2005), and in Borgs et al., Random subgraphs of finite graphs: II. The lace expansion and the triangle condition, Ann. Probab. (2005), the size of the largest connected component was estimated precisely for a large class of graphs including H(2,n) for $\vep\leq \Lambda V^{-1/3}$, where $\Lambda > 0$ is a constant and $V=n^2$ denotes the number of vertices in H(2,n). Until now, no matching lower bound on the size in the supercritical regime has been obtained. In this paper we prove that, when $\vep\gg (\log{V})^{1/3} V^{-1/3}$, then the largest connected component has size close to $2\vep V$ with high probability. We thus obtain a law of large numbers for the largest connected component size, and show that the corresponding values of $p$ are supercritical. Barring the factor $(\log{\chs{V}})^{1/3}$, this identifies the size of the largest connected component all the way down to the critical $p$ window. http://arxiv.org/abs/0801.1607 --------------------------------------------------------------- 6539. THE SECOND LARGEST COMPONENT IN THE SUPERCRITICAL 2D HAMMING GRAPH Malwina J. Luczak and Joel Spencer The 2-dimensional Hamming graph H(2,n) consists of the $n^2$ vertices $(i,j)$, $1\leq i,j\leq n$, two vertices being adjacent when they share a common coordinate. We examine random subgraphs of H(2,n) in percolation with edge probability $p$, so that the average degree $2(n-1)p=1+\epsilon$. Previous work by van der Hofstad and Luczak had shown that in the barely supercritical region $n^{-2/3}\ln^{1/3}n\ll \epsilon \ll 1$ the largest component has size $\sim 2\epsilon n$. Here we show that the second largest component has size close to $\epsilon^{-2}$, so that the dominant component has emerged. http://arxiv.org/abs/0801.1608 --------------------------------------------------------------- 6540. COMMENTS ON "REVERSE AUCTION: THE LOWEST POSITIVE INTEGER GAME" Adrian P. Flitney In Zeng et al. [Fluct. Noise Lett. 7 (2007) L439--L447] the analysis of the lowest unique positive integer game is simplified by some reasonable assumptions that make the problem tractable for arbitrary numbers of players. However, here we show that the solution obtained for rational players is not a Nash equilibrium and that a rational utility maximizer with full computational capability would arrive at a solution with a superior expected payoff. An exact solution is presented for the three- and four-player cases and an approximate solution for an arbitrary number of players. http://arxiv.org/abs/0801.1535 --------------------------------------------------------------- 6541. A GEOMETRIC PREFERENTIAL ATTACHMENT MODEL WITH FITNESS H. van den Esker We study a random graph $G_n$, which combines aspects of geometric random graphs and preferential attachment. The resulting random graphs have power-law degree sequences with finite mean and possibly infinite variance. In particular, the power-law exponent can be any value larger than 2. The vertices of $G_n$ are $n$ sequentially generated vertices chosen at random in the unit sphere in $\mathbb R^3$. A newly added vertex has $m $ edges attached to it and the endpoints of these edges are connected to old vertices or to the added vertex itself. The vertices are chosen with probability proportional to their current degree plus some initial attractiveness and multiplied by a function, depending on the geometry. http://arxiv.org/abs/0801.1612 --------------------------------------------------------------- 6542. ON EXCHANGEABLE RANDOM VARIABLES AND THE STATISTICS OF LARGE GRAPHS AND HYPERGRAPHS Tim D. Austin (UC and Los Angeles) De Finetti's classical result identifying the law of an exchangeable family of random variables as a mixture of i.i.d. laws was extended to structure theorems for more complex notions of exchangeability by Aldous, Hoover and Kallenberg. On the other hand, such exchangeable laws were first related to questions from combinatorics in an independent analysis by Fremlin and Talagrand, and again more recently in work of Tao, where they appear as a natural proxy for the `leading order statistics' of colourings of large graphs or hypergraphs. Moreover, this relation appears implicitly in the study of various more bespoke formalisms for handling `limit objects' of sequences of dense graphs or hypergraphs in a number of recent works. However, the connection between these works and the earlier probabilistic structural results seems to have gone largely unappreciated. In this survey we recall the basic results of the theory of exchangeable laws, and then explain the probabilistic versions of various interesting questions from graph and hypergraph theory that their connection motivates (particularly extremal questions on the testability of properties for graphs and hypergraphs). We also locate the notions of exchangeability of interest to us in the context of other classes of probability measures subject to various symmetries, in particular contrasting the methods employed to analyze exchangeable laws with related structural results in ergodic theory, particular the Furstenberg-Zimmer structure theorem for probability-preserving $\bbZ$- systems, which underpins Furstenberg's ergodic-theoretic proof of Szemer\'edi's Theorem. http://arxiv.org/abs/0801.1698 --------------------------------------------------------------- 6543. FREIDLIN-WENTZELL'S LARGE DEVIATIONS FOR STOCHASTIC EVOLUTION EQUATIONS Jiagang Ren and Xicheng Zhang We prove a Freidlin-Wentzell large deviation principle for general stochastic evolution equations with small perturbation multiplicative noises. In particular, our general result can be used to deal with a large class of quasi linear stochastic partial differential equations, such as stochastic porous medium equations and stochastic reaction diffusion equations with polynomial growth zero order term and $p$-Laplacian second order term. http://arxiv.org/abs/0801.1830 --------------------------------------------------------------- 6544. HARMONIC MEASURE AND SLE D. Beliaev and S. Smirnov In this paper we rigorously compute the average multifractal spectrum of harmonic measure on the boundary of SLE clusters. http://arxiv.org/abs/0801.1792 --------------------------------------------------------------- 6545. THE LEXICOGRAPHIC FIRST OCCURRENCE OF A I-II-III PATTERN Torey Burton and Anant P. Godbole and Brett M. Kindle Consider a random permutation $\pi\in{\cal S}_n$. In this paper, perhaps best classified as a contribution to discrete probability distribution theory, we study the {\it first} occurrence $X=X_n$ of a I-II-III-pattern, where "first" is interpreted in the lexicographic order induced by the 3-subsets of $[n]=\{1,2,...,n\}$. Of course if the permutation is I-II-III-avoiding then the first I-II-III-pattern never occurs, and thus $\e(X)=\infty$ for each $n$; to avoid this case, we also study the first occurrence of a I-II-III- pattern given a bijection $f:{\bf Z}^+\to{\bf Z}^+$. http://arxiv.org/abs/0801.1876 --------------------------------------------------------------- 6546. EFFECTIVE RESISTANCE OF RANDOM TREES Louigi Addario-Berry and Nicolas Broutin and Gabor Lugosi We investigate the effective resistance R_n and conductance C_n between the root and leaves a binary tree of height n. In this electrical network, the resistance of each edge e at distance d from the root is defined by r_e=2^d X_e where the X_e are i.i.d. positive random variables bounded away from zero and infinity. It is shown that E(R_n) = n*E(X_e) - (V(X_e)/E(X_e))*ln n + O(1) and V(R_n)=O(1). Some of the results are extended to the case when the underlying tree is a supercritical Galton--Watson tree. (In this case the correct scale for r_e is b^dX_e where b is the branching number of the tree.) http://arxiv.org/abs/0801.1909 --------------------------------------------------------------- 6547. POSITIVELY AND NEGATIVELY EXCITED RANDOM WALKS ON INTEGERS, WITH BRANCHING PROCESSES Elena Kosygina and Martin P.W. Zerner We consider excited random walks on the integers with a bounded number of i.i.d. cookies per site which may induce drifts both to the left and to the right. We extend the criteria for recurrence and transience by M. Zerner and for positivity of speed by A.-L. Basdevant and A. Singh to this case and also prove an annealed central limit theorem. The proofs are based on results from the literature concerning branching processes with migration and make use of a certain renewal structure. http://arxiv.org/abs/0801.1924 --------------------------------------------------------------- 6548. CONVEXITY AND SMOOTHNESS OF SCALE FUNCTIONS AND DE FINETTI'S CONTROL PROBLEM A. E. Kyprianou and V. Rivero and R. Song Motivated by a classical control problem from actuarial mathematics, we study smoothness and convexity properties of $q$-scale functions for spectrally negative L\'evy processes. Continuing from the very recent work of \cite{APP2007} and \cite{Loe} we strengthen their collective conclusions by showing, amongst other results, that whenever the L\'evy measure has a non-decreasing density which is log convex then for $q>0$ the scale function $W^{(q)}$ is convex on some half line $(a^*,\infty)$ where $a^*$ is the largest value at which $W^{(q)\prime}$ attains its global minimum. As a consequence we deduce that de Finetti's classical actuarial control problem is solved by a barrier strategy where the barrier is positioned at height $a^*$. http://arxiv.org/abs/0801.1951 --------------------------------------------------------------- 6549. GEOMETRIC GAMMA MAX-INFINITELY DIVISIBLE MODELS S. Satheesh and E. Sandhya A transformation of gamma max-infinitely divisible laws viz. geometric gamma max-infinitely divisible laws is considered in this paper. Some of its distributional and divisibility properties are discussed and a random time changed extremal process corresponding to this distribution is presented. A new kind of invariance (stability) under geometric maxima is proved and a max-AR(1) model corresponding to it is also discussed. http://arxiv.org/abs/0801.2083 --------------------------------------------------------------- 6550. THE LOCAL TIME OF THE CLASSICAL RISK PROCESS F. Cortes and J.A. Le\'on and J. Villa In this paper we give an explicit expression for the local time of the classical risk process and associate it with the density of an occupational measure. To do so, we approximate the local time by a suitable sequence of absolutely continuous random fields. Also, as an application, we analyze the mean of the times $s \in [0,T]$ such that $0\leq X_{s} \leq X_{s+ \epsilon} $ for some given $\epsilon>0$. http://arxiv.org/abs/0801.2106 --------------------------------------------------------------- 6551. Q-INVARIANT FUNCTIONS FOR SOME GENERALIZATIONS OF THE ORNSTEIN- UHLENBECK SEMIGROUP P. Patie We show that the multiplication operator associated to a fractional power of a Gamma random variable, with parameter q>0, maps the convex cone of the 1-invariant functions for a self-similar semigroup into the convex cone of the q-invariant functions for the associated Ornstein-Uhlenbeck (for short OU) semigroup. We also describe the harmonic functions for some other generalizations of the OU semigroup. Among the various applications, we characterize, through their Laplace transforms, the laws of first passage times above and overshoot for certain two-sided stable OU processes and also for spectrally negative semi-stable OU processes. These Laplace transforms are expressed in terms of a new family of power series which includes the generalized Mittag-Leffler functions. http://arxiv.org/abs/0801.2111 --------------------------------------------------------------- 6552. BOUNDS ON THE POINCARE CONSTANT OF ULTRA LOG-CONCAVE RANDOM VARIABLES Oliver Johnson We consider the discrete Poincar\'{e} constant, which relates the variance of a function to the expected square of its finite difference. We give an explicit bound on the Poincar\'{e} constant of ultra log-concave random variables in terms of their first two moments, and discuss how this bound relates to calculations performed by other authors. http://arxiv.org/abs/0801.2112 --------------------------------------------------------------- 6553. A STUDY OF COUNTS OF BERNOULLI STRINGS VIA CONDITIONAL POISSON PROCESSES Fred W. Huffer and Jayaram Sethuraman and Sunder Sethuraman We say that a string of length $d$ occurs, in a Bernoulli sequence, if a success is followed by exactly $(d-1)$ failures before the next success. The counts of such $d$-strings are of interest, and in specific independent Bernoulli sequences are known to correspond to asymptotic $d$-cycle counts in random permutations. In this note, we give a new framework, in terms of conditional Poisson processes, which allows for a quick characterization of the joint distribution of the counts of all $d$-strings, in a general class of Bernoulli sequences, as certain mixtures of the product of Poisson measures. This general class includes all Bernoulli sequences considered before, as well many new sequences. http://arxiv.org/abs/0801.2115 --------------------------------------------------------------- 6554. EXPONENTIAL BOUNDS IN THE LAW OF ITERATED LOGARITHM FOR MARTINGALES E. Ostrovsky and L.Sirota In this paper non-asymptotic exponential estimates are derived for tail of maximum martingale distribution by naturally norming in the spirit of the classical Law of Iterated Logarithm. Key words: Martingales, exponential estimations, moment, Banach spaces of random variables, tail of distribution, conditional expectation. http://arxiv.org/abs/0801.2125 --------------------------------------------------------------- 6555. FAR FIELD ASYMPTOTICS OF SOLUTIONS TO CONVECTION EQUATION WITH ANOMALOUS DIFFUSION Lorenzo Brandolese (ICJ) and Grzegorz Karch The initial value problem for the conservation law $\partial_t u+(-\Delta)^{\alpha/2}u+\nabla \cdot f(u)=0$ is studied for $\alpha\in (1,2)$ and under natural polynomial growth conditions imposed on the nonlinearity. We find the asymptotic expansion as $|x|\to \infty$ of solutions to this equation corresponding to initial conditions, decaying sufficiently fast at infinity. http://arxiv.org/abs/0801.1884 --------------------------------------------------------------- 6556. THE ORIGIN OF INFINITELY DIVISIBLE DISTRIBUTIONS: FROM DE FINETTI'S PROBLEM TO LEVY-KHINTCHINE FORMULA Francesco Mainardi and Sergei Rogosin The article provides an historical survey of the early contributions on infinitely divisible distributions starting from the pioneering works of de Finetti in 1929 up to the canonical forms developed in the thirties by Kolmogorov, Levy and Khintchine. Particular attention is paid to single out the personal contributions of the above authors that were published in Italian, French or Russian during the period 1929-1938. In Appendix we report the translation from the Russian into English of a fundamental paper by Khintchine published in Moscow in 1937. http://arxiv.org/abs/0801.1910 --------------------------------------------------------------- 6557. N-MONOTONE EXACT FUNCTIONALS Gert de Cooman and Matthias C. M. Troffaes and Enrique Miranda We study n-monotone functionals, which constitute a generalisation of n-monotone set functions. We investigate their relation to the concepts of exactness and natural extension, which generalise the notions of coherence and natural extension in the behavioural theory of imprecise probabilities. We improve upon a number of results in the literature, and prove among other things a representation result for exact n-monotone functionals in terms of Choquet integrals. http://arxiv.org/abs/0801.1962 --------------------------------------------------------------- 6558. A FUNCTIONAL CENTRAL LIMIT THEOREM FOR REGENERATIVE CHAINS G. Maillard and S. Sch\"opfer Using the regenerative scheme of Comets, Fern\'andez and Ferrari (2002), we establish a functional central limit theorem (FCLT) for discrete time stochastic processes (chains) with summable memory decay. Furthermore, under stronger assumptions on the memory decay, we identify the limiting variance in terms of the process only. As applications, we define classes of binary autoregressive processes and power-law Ising chains for which the FCLT is fulfilled. http://arxiv.org/abs/0801.2263 --------------------------------------------------------------- 6559. MALLIAVIN CALCULUS FOR DIFFERENCE APPROXIMATIONS OF MULTIDIMENSIONAL DIFFUSIONS: TRUNCATED LOCAL LIMIT THEOREM Alexey M. Kulik For a difference approximations of multidimensional diffusion, the truncated local limit theorem is proved. Under very mild conditions on the distribution of the difference terms, this theorem provides that the transition probabilities of these approximations, after truncation of some asymptotically negligible terms, possess a densities that converge uniformly to the transition probability density for the limiting diffusion and satisfy a uniform diffusion-type estimates. The proof is based on the new version of the Malliavin calculus for the product of finite family of measures, that may contain non-trivial singular components. An applications for uniform estimates for mixing and convergence rates for difference approximations to SDE's and for convergence of difference approximations for local times of multidimensional diffusions are given. http://arxiv.org/abs/0801.2319 --------------------------------------------------------------- 6560. ANALYSIS OF THE STOCHASTIC FITZHUGH-NAGUMO SYSTEM Stefano Bonaccorsi and Elisa Mastrogiacomo In this paper we study a system of stochastic differential equations with dissipative nonlinearity which arise in certain neurobiology models. Besides proving existence, uniqueness and continuous dependence on the initial datum, we shall be mainly concerned with the asymptotic behaviour of the solution. We prove the existence of an invariant ergodic measure $\nu$ associated with the transition semigroup $P_t$; further, we identify its infinitesimal generator in the space $L^2(H;\nu)$. http://arxiv.org/abs/0801.2325 --------------------------------------------------------------- 6561. THE EINSTEIN RELATION FOR RANDOM WALKS ON GRAPHS Andras Telcs This paper investigates the Einstein relation; the connection between the volume growth, the resistance growth and the expected time a random walk needs to leave a ball on a weighted graph. The Einstein relation is proved under different set of conditions. In the simplest case it is shown under the volume doubling and time comparison principles. This and the other set of conditions provide the basic framework for the study of (sub-) diffusive behavior of the random walks on weighted graphs. http://arxiv.org/abs/0801.2336 --------------------------------------------------------------- 6562. UPPER BOUNDS FOR TRANSITION PROBABILITIES ON GRAPHS AND ISOPERIMETRIC INEQUALITIES Andras Telcs In this paper necessary and sufficient conditions are presented for heat kernel upper bounds for random walks on weighted graphs. Several equivalent conditions are given in the form of isoperimetric inequalities. http://arxiv.org/abs/0801.2341 --------------------------------------------------------------- 6563. RANDOM WALKS ON GRAPHS WITH VOLUME AND TIME DOUBLING Andras Telcs This paper studies the on- and off-diagonal upper estimate and the two- sided transition probability estimate of random walks on weighted graphs. http://arxiv.org/abs/0801.2351 --------------------------------------------------------------- 6564. ON WASSERSTEIN GEOMETRY OF THE SPACE OF GAUSSIAN MEASURES Asuka Takatsu The space which consists of measures having finite second moment is an infinite dimensional metric space endowed with Wasserstein distance, while the space of Gaussian measures on Euclidean space is parameterized by mean and covariance matrices, hence a finite dimensional manifold. By restricting to the space of Gaussian measures inside the space of probability measures, we manege to provide detailed descriptions of the Wasserstein geometry from a Riemannian geometric viewpoint. In particular, using the results from the Monge- Kantrovich transport theory, an explicit expression of geodesics interpolating two Gaussian measures. It follows that the space of Gaussian measures is geodesically convex in the space of probability measures. Also, a Riemannian metric which induces the Wasserstein distance is specified. Using the Riemannian metric, a formula for the sectional curvatures of the space of Gaussian measures on the plane is written out in terms of the eigenvalues of the covariance matrix. http://arxiv.org/abs/0801.2250 --------------------------------------------------------------- 6565. THE VOLUME AND TIME COMPARISON PRINCIPLE AND TRANSITION PROBABILITY ESTIMATES FOR RANDOM WALKS Andras Telcs This paper presents necessary and sufficient conditions for on- and off-diagonal transition probability estimates for random walks on weighted graphs. On the integer lattice and on may fractal type graphs both the volume of a ball and the mean exit time from a ball is independent of the centre, uniform in space. Here the upper estimate is given without such restriction and two-sided estimate is given if uniformity in the space assumed only for the mean exit time. http://arxiv.org/abs/0801.2393 --------------------------------------------------------------- 6566. STOCHASTIC POROUS MEDIA EQUATION AND SELF-ORGANIZED CRITICALITY Viorel Barbu (Institute of Mathematics "Octav Mayer" and Iasi and Romania) and Giuseppe Da Prato (Scuola Normale Superiore di Pisa, Italy) and Michael R\"ockner (Faculty of Mathematics, Bielefeld, Germany and Departments of Mathematics and Statistics, Purdue University, USA) The existence and uniqueness of nonnegative strong solutions for stochastic porous media equations with noncoercive monotone diffusivity function and Wiener forcing term is proven. The finite time extinction of solutions with high probability is also proven in 1-D. The results are relevant for self-organized critical behaviour of stochastic nonlinear diffusion equations with critical states. http://arxiv.org/abs/0801.2478 --------------------------------------------------------------- 6567. BAXTER'S INEQUALITY FOR FRACTIONAL BROWNIAN MOTION-TYPE PROCESSES WITH HURST INDEX LESS THAN 1/2 Akihiko Inoue and Yukio Kasahara and Punam Phartyal The aim of this paper is to prove an analogue of Baxter's inequality for fractional Brownian motion-type processes with Hurst index less than 1/2. This inequality is concerned with the norm estimate of the difference between finite- and infinite-past predictor coefficients. http://arxiv.org/abs/0801.2509 --------------------------------------------------------------- 6568. THERMODYNAMIC LIMIT FOR THE INVARIANT MEASURES IN SUPERCRITICAL ZERO RANGE PROCESSES In\'es Armend\'ariz and Michail Loulakis We prove a strong form of the equivalence of ensembles for the invariant measures of zero range processes conditioned to a supercritical density of particles. It is known that in this case there is a single site that accomodates a macroscopically large number of the particles in the system. We show that in the thermodynamic limit the rest of the sites have joint distribution equal to the grand canonical measure at the critical density. This improves the result of Gro\ss kinsky, Sch\"{u}tz and Spohn, where convergence is obtained for the finite dimensional marginals. We obtain as corollaries limit theorems for the order statistics of the components and for the fluctuations of the bulk. http://arxiv.org/abs/0801.2511 --------------------------------------------------------------- 6569. THE VARIANCE OF THE SHOCK IN THE HAD PROCESS Cristian F. Coletti and Pablo A. Ferrari and Leandro P.R. Pimentel We consider the Hammersley-Aldous-Diaconis (HAD) process with sinks and sources such that there is a microscopic shock at every time $t$; denote $Z(t)$ its position. We show that the mean and variance of $Z(t)$ are linear functions of $t$ and compute explicitely the respective constants in function of the left and right densities. Furthermore, we describe the dependence of $Z(t)$ on the initial configuration in the scale $\sqrt t$ and, as a corollary, prove a central limit theorem. http://arxiv.org/abs/0801.2526 --------------------------------------------------------------- 6570. RECURRENCE TIMES AND LARGE DEVIATIONS Yong Moo Chung We give a criterion to determine the large deviation rate functions for abstract dynamical systems on towers. As an application of this criterion we show the level 2 large deviation principle for some class of smooth interval maps with nonuniform hyperbolicity. http://arxiv.org/abs/0801.2409 --------------------------------------------------------------- 6571. UNIFORMLY SPREAD MEASURES AND VECTOR FIELDS Mikhail Sodin and Boris Tsirelson We show that two different ideas of uniform spreading of locally finite measures in the d-dimensional Euclidean space are equivalent. The first idea is formulated in terms of finite distance transportations to the Lebesgue measure, while the second idea is formulated in terms of vector fields connecting a given measure with the Lebesgue measure. http://arxiv.org/abs/0801.2505 --------------------------------------------------------------- 6572. EDGEWORTH EXPANSION OF THE LARGEST EIGENVALUE DISTRIBUTION FUNCTION OF GOE Leonard N. Choup In this paper we focus on the large n probability distribution function of the largest eigenvalue in the Gaussian Orthogonal Ensemble of n by n matrices (GOEn). We prove an Edgeworth type Theorem for the largest eigenvalue probability distribution function of GOEn. The correction terms to the limiting probability distribution are expressed in terms of the same Painleve II functions appearing in the Tracy-Widom distribution. We conclude with a brief discussion of the GSEn case. http://arxiv.org/abs/0801.2620 --------------------------------------------------------------- 6573. TOTAL-VARIATION CUTOFF IN BIRTH-AND-DEATH CHAINS Jian Ding and Eyal Lubetzky and Yuval Peres The cutoff phenomenon describes a case where a Markov chain exhibits a sharp transition in its convergence to stationarity. In 1996, Diaconis surveyed this phenomenon, and asked how one could recognize its occurrence in families of finite ergodic Markov chains. In 2004, the third author noted that a necessary condition for cutoff in a family of reversible chains is that the product of the mixing-time and spectral-gap tends to infinity, and conjectured that in many settings, this condition should also be sufficient. Diaconis and Saloff-Coste (2006) verified this conjecture for birth-and-death chains with the convergence measured in separation. It is natural to ask whether the conjecture holds for these chains in the more widely used total- variation distance. In this work, we confirm the above conjecture for all continuous- time or lazy discrete-time birth-and-death chains, with convergence measured via total-variation distance. Namely, if the product of the mixing-time and spectral-gap tends to infinity, the chains exhibit cutoff at the maximal hitting time of the stationary distribution median, with a window of at most the geometric mean between the relaxation-time and mixing-time. http://arxiv.org/abs/0801.2625 --------------------------------------------------------------- 6574. HARNACK INEQUALITY AND APPLICATIONS FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS Feng-Yu Wang and Chenggui Yuan Gradient estimates and a Harnack inequality are established for the semigroup associated to stochastic differential equations driven by Poisson processes. As applications, estimates of the transition probability density, the compactness and ultraboundedness of the semigroup are studied in terms of the corresponding invariant measure. http://arxiv.org/abs/0801.2668 --------------------------------------------------------------- 6575. CONSTRUCTION OF AN EDWARDS' PROBABILITY MEASURE ON $\MATHCAL{C} (\MATHBB{R}_+, \MATHBB{R})$ Joseph Najnudel In this article, we prove that the measures $\mathbb{Q}_T$ associated to the one-dimensional Edwards' model on the interval $[0,T]$ converge to a limit measure $\mathbb{Q}$ when $T$ goes to infinity, in the following sense : for all $s \geq 0$ and for all events $\Lambda_s$ depending on the canonical process only up to time $s$, $\mathbb{Q}_T (\Lambda_s) \to \mathbb{Q} (\Lambda_s)$. Moreover, we prove that, if $\mathbb{P}$ is Wiener measure, there exists a martingale $(D_s)_{s \in \mathbb{R}_+}$ such that $ \mathbb{Q} (\Lambda_s) = \mathbb{E}_{\mathbb{P}} (\mathds{1}_{\Lambda_s} D_s)$, and we give an explicit expression for this martingale. http://arxiv.org/abs/0801.2751 --------------------------------------------------------------- 6576. DISCRETE APPROXIMATION OF A STABLE SELF-SIMILAR STATIONARY INCREMENTS PROCESS Cl\'ement Dombry (LMA) and Nadine Guillotin-Plantard (ICJ) The aim of this paper is to present a result of discrete approximation of some class of stable self-similar stationary increments processes. The properties of such processes were intensively investigated, but little is known on the context in which such processes can arise. To our knowledge, discretisation and convergence theorems are available only in the case of stable L\'evy motions and fractional Brownian motions. This paper yields new results in this direction. Our main result is the convergence of the random rewards schema, which was firstly introduced by Cohen and Samorodnitsky, and that we consider in a more general setting. Strong relationships with Kesten and Spitzer's random walk in random sceneries are evidenced. Finally, we study some path properties of the limit process. http://arxiv.org/abs/0801.2753 --------------------------------------------------------------- 6577. INTEGRABILITY OF EXIT TIMES AND BALLISTICITY FOR RANDOM WALKS IN DIRICHLET ENVIRONMENT Laurent Tournier (ICJ) We consider random walks in Dirichlet environment, introduced by Enriquez and Sabot in 2006. As this distribution on environments is not uniformly elliptic, the annealed integrability of exit times out of a given finite subset is a non-trivial property. We provide here an explicit equivalent condition for this integrability to happen, on general directed graphs. Such integrability problems arise for instance from the definition of Kalikow auxiliary random walk. Using our condition, we prove a refined version of the ballisticity criterion given by Enriquez and Sabot. http://arxiv.org/abs/0801.2875 --------------------------------------------------------------- 6578. ON BESOV REGULARITY OF BROWNIAN MOTIONS IN INFINITE DIMENSIONS Tuomas Hytonen and Mark Veraar We extend to the vector-valued situation some earlier work of Ciesielski and Roynette on the Besov regularity of the paths of the classical Brownian motion. We also consider a Brownian motion as a Besov space valued random variable. It turns out that a Brownian motion, in this interpretation, is a Gaussian random variable with some pathological properties. We prove estimates for the first moment of the Besov norm of a Brownian motion. To obtain such results we estimate expressions of the form $\E \sup_{n\geq 1}\|\xi_n\|$, where the $\xi_n$ are independent centered Gaussian random variables with values in a Banach space. Using isoperimetric inequalities we obtain two-sided inequalities in terms of the first moments and the weak variances of $\xi_n$. http://arxiv.org/abs/0801.2959 --------------------------------------------------------------- 6579. EVERY MINOR-CLOSED PROPERTY OF SPARSE GRAPHS IS TESTABLE Itai Benjamini and Oded Schramm and Asaf Shapira Testing a property $P$ of graphs in the bounded degree model deals with the following problem: given a graph $G$ of bounded degree $d$ we should distinguish (with probability 0.9, say) between the case that $G$ satisfies $P$ and the case that one should add/remove at least $\epsilon d n$ edges of $G$ to make it satisfy $P$. In sharp contrast to property testing of dense graphs, which is relatively well understood, very few properties are known to be testable in bounded degree graphs with a constant number of queries. In this paper we identify for the first time a large (and natural) family of properties that can be efficiently tested in bounded degree graphs, by showing that every minor-closed graph property can be tested with a constant number of queries. As a special case, we infer that many well studied graph properties, like being planar, outer-planar, series-parallel, bounded genus, bounded tree-width and several others, are testable with a constant number of queries. None of these properties was previously known to be testable even with $o(n)$ queries. The proof combines results from the theory of graph minors with results on convergent sequences of sparse graphs, which rely on martingale arguments. http://arxiv.org/abs/0801.2797 --------------------------------------------------------------- 6580. MALLIAVIN CALCULUS AND DECOUPLING INEQUALITIES IN BANACH SPACES Jan Maas We develop a theory of Malliavin calculus for Banach space valued random variables. Using radonifying operators instead of symmetric tensor products we extend the Wiener-Ito isometry to Banach spaces. In the white noise case we obtain two sided L^p-estimates for multiple stochastic integrals in arbitrary Banach spaces. It is shown that the Malliavin derivative is bounded on vector-valued Wiener-Ito chaoses. Our main tools are decoupling inequalities for vector-valued random variables. In the opposite direction we use Meyer's inequalities to give a new proof of a decoupling result for Gaussian chaoses in UMD Banach spaces. http://arxiv.org/abs/0801.2899 --------------------------------------------------------------- 6581. $C^1$-GENERIC SYMPLECTIC DIFFEOMORPHISMS: PARTIAL HYPERBOLICITY AND LYAPUNOV EXPONENTS Jairo Bochi It is proven that for a $C^1$-generic symplectic diffeomorphism $f$ of any closed manifold, the Oseledets splitting along almost every orbit is either trivial or partially hyperbolic. In addition, if $f$ is not Anosov then all the exponents in the center bundle vanish. This establishes in full a result announced by Ma\~n\'e in the ICM 1983. The main technical novelty is a probabilistic method for the construction of perturbations (using random walks). http://arxiv.org/abs/0801.2960 --------------------------------------------------------------- 6582. A NEW CONCEPT OF STRONG CONTROLLABILITY VIA THE SCHUR COMPLEMENT IN ADAPTIVE TRACKING Bernard Bercu and Victor Vazquez We propose a new concept of strong controllability associated with the Schur complement of a suitable limiting matrix. This concept allows us to extend the previous results associated with multidimensional ARX models. On the one hand, we carry out a sharp analysis of the almost sure convergence for both least squares and weighted least squares algorithms. On the other hand, we also provide a central limit theorem and a law of iterated logarithm for these two stochastic algorithms. Our asymptotic results are illustrated by numerical simulations. http://arxiv.org/abs/0801.2991 --------------------------------------------------------------- 6583. ESTIMATION OF QUADRATIC VARIATION FOR TWO-PARAMETER DIFFUSIONS Anthony R\'eveillac In this paper we give a central limit theorem for the weighted quadratic variations process of a two-parameter Brownian motion. As an application, we show that the discretized quadratic variations $\sum_{i=1}^{[n s]} \sum_{j=1}^{[n t]} | \Delta_{i,j} Y |^2$ of a two-parameter diffusion $Y=(Y_{(s,t)})_{(s,t)\in[0,1]^2}$ observed on a regular grid $G_n$ is an asymptotically normal estimator of the quadratic variation of $Y$ as $n $ goes to infinity. http://arxiv.org/abs/0801.3027 --------------------------------------------------------------- 6584. A LOWER BOUND FOR THE CHUNG-DIACONIS-GRAHAM RANDOM PROCESS Martin Hildebrand Chung, Diaconis, and Graham considered random processes of the form X_{n+1}=a_n X_n+b_n (mod p) where p is odd, X_0=0, a_n=2 always, and b_n are i.i.d. for n=0,1,2,... . In this paper, we show that if P(b_n=-1)=P{b_n=0)=P(b_n=1)=1/3, then there exists a constant c>1 such that c log_2 p steps are not enough to make X_n get close to uniformly distributed on the integers mod p. http://arxiv.org/abs/0801.3094 --------------------------------------------------------------- 6585. APPROXIMATE WORD MATCHES BETWEEN TWO RANDOM SEQUENCES Conrad J. Burden and Miriam R. Kantorovitz and Susan R. Wilson Given two sequences over a finite alphabet $\mathcal{L}$, the $D_2$ statistic is the number of $m$-letter word matches between the two sequences. This statistic is used in bioinformatics for expressed sequence tag database searches. Here we study a generalization of the $D_2$ statistic in the context of DNA sequences, under the assumption of strand symmetric Bernoulli text. For $k0$ on $\{x_i=0\}$. The special case $d=2$, $b_i=\theta_i-x_i$ is required in work of Dawson-Greven-den Hollander-Sun-Swart on mean fields limits of block averages for 2-type branching models on a hierarchical group. The proofs make use of some new methods, including Cotlar's lemma to establish asymptotic orthogonality of the derivatives of an associated semigroup at different times,and a refined integration by parts technique from Dawson- Perkins]. As a by-product of the proof we obtain the strong Feller property of the associated resolvent. http://arxiv.org/abs/0801.3257 --------------------------------------------------------------- 6591. A NOTE ABOUT CONDITIONAL ORNSTEIN-UHLENBECK PROCESSES Amel Bentata (PMA) In this short note, the identity in law, which was obtained by P. Salminen, between on one hand, the Ornstein-Uhlenbeck process with parameter gamma, killed when it reaches 0, and on the other hand, the 3-dimensional radial Ornstein-Uhlenbeck process killed exponentially at rate gamma and conditioned to hit 0, is derived from a simple absolute continuity relationship. http://arxiv.org/abs/0801.3261 --------------------------------------------------------------- 6592. THE EXECUTION GAME Ciamac C. Moallemi and Beomsoo Park and Benjamin Van Roy We consider a trader who aims to liquidate a large position in the presence of an arbitrageur who hopes to profit from the trader's activity. The arbitrageur is uncertain about the trader's position and learns from observed market activity. This is a dynamic game with asymmetric information. We present an algorithm for computing perfect Bayesian equilibrium behavior and conduct numerical experiments. Our results demonstrate that the trader's strategy differs in important ways from one that would be optimal in the absence of an arbitrageur. In particular, the trader's actions depend on and influence the arbitrageur's beliefs. Accounting for the presence of a strategic adversary can greatly reduce transaction costs. http://arxiv.org/abs/0801.3001 --------------------------------------------------------------- 6593. POISSON SUSPENSIONS AND ENTROPY FOR INFINITE TRANSFORMATIONS Elise Janvresse and Tom Meyerovitch and Emmanuel Roy and Thierry De La Rue The Poisson entropy of an infinite-measure-preserving transformation is defined as the Kolmogorov entropy of its Poisson suspension. In this article, we relate Poisson entropy with other definitions of entropy for infinite transformations: For quasi-finite transformations we prove that Poisson entropy coincides with Krengel's and Parry's entropy. In particular, this implies that for null-recurrent Markov chains, the usual formula for the entropy $- \sum q_i p_{i,j}\log p_{i,j}$ holds in any of the definitions for entropy. Poisson entropy dominates Parry's entropy in any conservative transformation. We also prove that relative entropy (in the sense of Danilenko and Rudolph) coincides with the relative Poisson entropy. Thus, for any factor of a conservative transformation, difference of the Krengel's entropy is equal to the difference of the Poisson entropies. Finally, we prove the existence of a maximal (Pinsker) factor with zero (Poisson, Krengel, Parry) entropy for quasi- finite transformations. This answers affirmatively the question about existence of a Pinsker factor in the sense of Krengel for quasi-finite transformations, a question raised in arXiv:0705.2148v3. http://arxiv.org/abs/0801.3155 --------------------------------------------------------------- 6594. OCCUPATION DENSITIES FOR CERTAIN PROCESSES RELATED TO FRACTIONAL BROWNIAN MOTION Khalifa Es-Sebaiy and David Nualart and Youssef Ouknine and Ciprian Tudor (CES and SAMOS) In this paper we establish the existence of a square integrable occupation density for two classes of stochastic processes. First we consider a Gaussian process with an absolutely continuous random drift, and secondly we handle the case of a (Skorohod) integral with respect to the fractional Brownian motion with Hurst parameter $H>\frac 12$. The proof of these results uses a general criterion for the existence of a square integrable local time, which is based on the techniques of Malliavin calculus. http://arxiv.org/abs/0801.3314 --------------------------------------------------------------- 6595. THE LINEAGE PROCESS IN GALTON--WATSON TREES AND GLOBALLY CENTERED DISCRETE SNAKES Jean-Fran\c{c}ois Marckert We consider branching random walks built on Galton--Watson trees with offspring distribution having a bounded support, conditioned to have $n $ nodes, and their rescaled convergences to the Brownian snake. We exhibit a notion of ``globally centered discrete snake'' that extends the usual settings in which the displacements are supposed centered. We show that under some additional moment conditions, when $n$ goes to $+\infty$, ``globally centered discrete snakes'' converge to the Brownian snake. The proof relies on a precise study of the lineage of the nodes in a Galton--Watson tree conditioned by the size, and their links with a multinomial process [the lineage of a node $u$ is the vector indexed by $(k,j)$ giving the number of ancestors of $u$ having $k$ children and for which $u$ is a descendant of the $j$th one]. Some consequences concerning Galton--Watson trees conditioned by the size are also derived. http://arxiv.org/abs/0801.3330 --------------------------------------------------------------- 6596. CONVEXITY, TRANSLATION INVARIANCE AND SUBADDITIVITY FOR $G$- EXPECTATIONS AND RELATED RISK MEASURES Long Jiang Under the continuous assumption on the generator $g$, Briand et al. [Electron. Comm. Probab. 5 (2000) 101--117] showed some connections between $g$ and the conditional $g$-expectation $({\mathcal{E}}_g[\cdot|{\mathcal{F}}_t])_{t\in[0,T]}$ and Rosazza Gianin [Insurance: Math. Econ. 39 (2006) 19--34] showed some connections between $g$ and the corresponding dynamic risk measure $(\rho^g_t)_{t\in[0,T]}$. In this paper we prove that, without the additional continuous assumption on $g $, a $g$-expectation ${\mathcal{E}}_g$ satisfies translation invariance if and only if $g$ is independent of $y$, and ${\mathcal{E}}_g$ satisfies convexity (resp. subadditivity) if and only if $g$ is independent of $y$ and $g$ is convex (resp. subadditive) with respect to $z$. By these conclusions we deduce that the static risk measure $\rho^g$ induced by a $g$-expectation $ {\mathcal{E}}_g$ is a convex (resp. coherent) risk measure if and only if $g$ is independent of $y$ and $g$ is convex (resp. sublinear) with respect to $z$. Our results extend the results in Briand et al. [Electron. Comm. Probab. 5 (2000) 101--117] and Rosazza Gianin [Insurance: Math. Econ. 39 (2006) 19--34] on these subjects. http://arxiv.org/abs/0801.3340 --------------------------------------------------------------- 6597. EVOLUTIONARILY STABLE STRATEGIES OF RANDOM GAMES, AND THE VERTICES OF RANDOM POLYGONS Sergiu Hart and Yosef Rinott and Benjamin Weiss An evolutionarily stable strategy (ESS) is an equilibrium strategy that is immune to invasions by rare alternative (``mutant'') strategies. Unlike Nash equilibria, ESS do not always exist in finite games. In this paper we address the question of what happens when the size of the game increases: does an ESS exist for ``almost every large'' game? Letting the entries in the $n \times n$ game matrix be independently randomly chosen according to a distribution $F$, we study the number of ESS with support of size $2.$ In particular, we show that, as $n\to \infty$, the probability of having such an ESS: (i) converges to 1 for distributions $F$ with ``exponential and faster decreasing tails'' (e.g., uniform, normal, exponential); and (ii) converges to $1-1/\sqrt{e}$ for distributions $F$ with ``slower than exponential decreasing tails'' (e.g., lognormal, Pareto, Cauchy). Our results also imply that the expected number of vertices of the convex hull of $n$ random points in the plane converges to infinity for the distributions in (i), and to 4 for the distributions in (ii). http://arxiv.org/abs/0801.3353 --------------------------------------------------------------- 6598. ONE-DIMENSIONAL STEPPING STONE MODELS, SARDINE GENETICS AND BROWNIAN LOCAL TIME Richard Durrett and Mateo Restrepo Consider a one-dimensional stepping stone model with colonies of size $M$ and per-generation migration probability $\nu$, or a voter model on $ \mathbb{Z}$ in which interactions occur over a distance of order $K$. Sample one individual at the origin and one at $L$. We show that if $M\nu/L$ and $L/K^2$ converge to positive finite limits, then the genealogy of the sample converges to a pair of Brownian motions that coalesce after the local time of their difference exceeds an independent exponentially distributed random variable. The computation of the distribution of the coalescence time leads to a one-dimensional parabolic differential equation with an interesting boundary condition at 0. http://arxiv.org/abs/0801.3370 --------------------------------------------------------------- 6599. CONVERGENCE OF FINITE-DIMENSIONAL LAWS OF THE WEIGHTED QUADRATIC VARIATIONS PROCESS FOR SOME FRACTIONAL BROWNIAN SHEETS Anthony Reveillac In this paper we state and prove a central limit theorem for the finite-dimensional laws of the quadratic variations process of certain fractional Brownian sheets. The main tool of this article is a method developed by Nourdin and Nualart based on the Malliavin calculus. http://arxiv.org/abs/0801.3416 --------------------------------------------------------------- 6600. QUENCHED CONVERGENCE OF A SEQUENCE OF SUPERPROCESSES IN R^D AMONG POISSONIAN OBSTACLES Amandine Veber We prove a convergence theorem for a sequence of super-Brownian motions moving among hard Poissonian obstacles, when the intensity of the obstacles grows to infinity but their diameters shrink to zero in an appropriate manner. The superprocesses are shown to converge in probability for the law $\mathbf{P}$ of the obstacles, and $\mathbf{P}$-almost surely for a subsequence, towards a superprocess with underlying spatial motion given by Brownian motion and (inhomogeneous) branching mechanism $\psi(u,x)$ of the form $\psi(u,x)= u^2+ \kappa(x)u$, where $\kappa(x)$ depends on the density of the obstacles. This work draws on similar questions for a single Brownian motion. In the course of the proof, we establish precise estimates for integrals of functions over the Wiener sausage, which are of independent interest. http://arxiv.org/abs/0801.3444 --------------------------------------------------------------- 6601. ON THE CONDENSED DENSITY OF THE GENERALIZED EIGENVALUES OF PENCILS OF HANKEL GAUSSIAN RANDOM MATRICES AND APPLICATIONS Piero Barone Pencils of Hankel matrices whose elements have a joint Gaussian distribution with nonzero mean and not identical covariance are considered. An approximation to the distribution of the squared modulus of their determinant is computed which allows to get a closed form approximation of the condensed density of the generalized eigenvalues of the pencils. Implications of this result for solving several moments problems are discussed and some numerical examples are provided. http://arxiv.org/abs/0801.3352 --------------------------------------------------------------- 6602. STATISTICAL ARBITRAGE AND OPTIMAL TRADING WITH TRANSACTION COSTS IN FUTURES MARKETS Theodoros Tsagaris We consider the Brownian market model and the problem of expected utility maximization of terminal wealth. We, specifically, examine the problem of maximizing the utility of terminal wealth under the presence of transaction costs of a fund/agent investing in futures markets. We offer some preliminary remarks about statistical arbitrage strategies and we set the framework for futures markets, and introduce concepts such as margin, gearing and slippage. The setting is of discrete time, and the price evolution of the futures prices is modelled as discrete random sequence involving Ito's sums. We assume the drift and the Brownian motion driving the return process are non- observable and the transaction costs are represented by the bid-ask spread. We provide explicit solution to the optimal portfolio process, and we offer an example using logarithmic utility. http://arxiv.org/abs/0801.3348 --------------------------------------------------------------- 6603. HARMONIC ANALYSIS OF STOCHASTIC EQUATIONS AND BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS Freddy Delbaen and Shanjian Tang The BMO martingale theory is extensively used to study nonlinear multi-dimensional stochastic equations (SEs) in $\cR^p$ ($p\in [1, \infty)$) and backward stochastic differential equations (BSDEs) in $\cR^p\times \cH^p$ ($p\in (1, \infty)$) and in $\cR^\infty\times \bar{\cH^\infty}^{BMO}$, with the coefficients being allowed to be unbounded. In particular, the probabilistic version of Fefferman's inequality plays a crucial role in the development of our theory, which seems to be new. Several new results are consequently obtained. The particular multi-dimensional linear case for SDEs and BSDEs are separately investigated, and the existence and uniqueness of a solution is connected to the property that the elementary solutions-matrix for the associated homogeneous SDE satisfies the reverse H\"older inequality for some suitable exponent $p\ge 1$. Finally, we establish some relations between Kazamaki's quadratic critical exponent $b(M)$ of a BMO martingale $M$ and the spectral radius of the solution operator for the $M$-driven SDE, which lead to a characterization of Kazamaki's quadratic critical exponent of BMO martingales being infinite. http://arxiv.org/abs/0801.3505 --------------------------------------------------------------- 6604. MAJORIZING MEASURES AND PROPORTIONAL SUBSETS OF BOUNDED ORTHONORMAL SYSTEMS Olivier Guedon and Shahar Mendelson and Alain Pajor and Nicole Tomczak-Jaegermann In this article we prove that for any orthonormal system $ (\vphi_j)_{j=1}^n \subset L_2$ that is bounded in $L_{\infty}$, and any $1 < k 0$, where $S^\circ $ is the subset of all matrices which have strictly positive entries. We state conditions on the distribution of the random matrix $X_1$ which ensure that the logarithms of the entries, of the norm, and of the spectral radius of the products $X^{(n)}$, $n\ge 1$, are in the domain of attraction of a stable law. http://arxiv.org/abs/0801.3780 --------------------------------------------------------------- 6608. NOVEL BOUNDS ON MARGINAL PROBABILITIES Joris M. Mooij and Hilbert J. Kappen We derive two related novel bounds on single-variable marginal probability distributions in factor graphs with discrete variables. The first method propagates bounds over a subtree of the factor graph rooted in the variable, and the second method propagates bounds over the self-avoiding walk tree starting at the variable. By construction, both methods not only bound the exact marginal probability distribution of a variable, but also its approximate Belief Propagation marginal (``belief''). Thus, apart from providing a practical means to calculate bounds on marginals, our contribution also lies in an increased understanding of the error made by Belief Propagation. Empirically, we show that our bounds often outperform existing bounds in terms of accuracy and/or computation time. We also show that our bounds can yield nontrivial results for medical diagnosis inference problems. http://arxiv.org/abs/0801.3797 --------------------------------------------------------------- 6609. CONSTRUCTION AND UNIQUENESS FOR REFLECTED BSDE UNDER LINEAR INCREASING CONDITION G. Jia and Mingyu Xu In this paper, we study the uniqueness of the solution of reflected BSDE with one or two barriers, under continuous and linear increasing condition of generator $g$. Before that we study the construction of solution of of reflected BSDE with one or two barriers. http://arxiv.org/abs/0801.3718 --------------------------------------------------------------- 6610. ALGORITHM FOR SOLVING OPTIMIZATION PROBLEMS WITH INTERVAL VALUED PROBABILITY MEASURE Phantipa Thipwiwatpotjana and Weldon A. Lodwick We are concerned with three types of uncertainties: probabilistic, possibilitistic and interval. By using possibility and necessity measures as an Interval Valued Probability Measure (IVPM), we present IVPM's interval expected values whose possibility distributions are in the form of polynomials. By working with interval expected values of independent uncertainty coefficients in a linear optimization problem together with operations suggested in Lodwick and Jamison (2007), the problem after applying these operations becomes a linear programming problem with constant coefficients. This is achieved by the application of two functions. The first is applied to the interval coefficients, v: I -> R^k, where I= {[a,b] | a <= b}. The second is u: R^k -> R, applied to the product we got from a previous function. Similar concepts hold for any types of optimization problems with linear constraints. Moreover, it implied that optimization problems containing all three types of uncertainties in one problem can be solved as ordinary optimization problems. http://arxiv.org/abs/0801.3816 --------------------------------------------------------------- 6611. SMOOTH SOLUTIONS OF NON-LINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS Xicheng Zhang In this paper, we study the regularities of solutions of nonlinear stochastic partial differential equations in the framework of Hilbert scales. Then we apply our general result to several typical nonlinear SPDEs such as stochastic Burgers and Ginzburg-Landau's equations on the real line, stochastic 2D Navier-Stokes equations in the whole space and a stochastic tamed 3D Navier-Stokes equation in the whole space, and obtain the existence of their respectively smooth solutions. http://arxiv.org/abs/0801.3883 --------------------------------------------------------------- 6612. AN LQ PROBLEM FOR THE HEAT EQUATION ON THE HALFLINE WITH DIRICHLET BOUNDARY CONTROL AND NOISE G. Fabbri and B. Goldys A linear quadratic problem for a system governed by a heat equation with a Dirichlet boundary control and a Dirichlet boundary noise on halfline is studied. To this end the problem is reformulated as a stochastic evolution equation in a certain weighted L2 space. An appropriate choice of weight allows us to prove a stronger regularity for the boundary terms appearing in the infinite dimensional state equation. The direct solution of the Riccati equation related to the associated non-stochastic problem is used to find the solution of the problem in feedback form and to write the value function of the problem. http://arxiv.org/abs/0801.3888 --------------------------------------------------------------- 6613. EXIT PROBLEMS RELATED TO THE PERSISTENCE OF SOLITONS FOR THE KORTEWEG-DE VRIES EQUATION WITH SMALL NOISE Anne De Bouard (CMAP) and Eric Gautier (CREST) We consider two exit problems for the Korteweg-de Vries equation perturbed by an additive white in time and colored in space noise of amplitude a. The initial datum gives rise to a soliton when a=0. It has been proved recently that the solution remains in a neighborhood of a randomly modulated soliton for times at least of the order of a^{-2}. We prove exponential upper and lower bounds for the small noise limit of the probability that the exit time from a neighborhood of this randomly modulated soliton is less than T, of the same order in a and T. We obtain that the time scale is exactly the right one. We also study the similar probability for the exit from a neighborhood of the deterministic soliton solution. We are able to quantify the gain of eliminating the secular modes to better describe the persistence of the soliton. http://arxiv.org/abs/0801.3894 --------------------------------------------------------------- 6614. ON LARGE INTERSECTION AND SELF-INTERSECTION LOCAL TIMES IN DIMENSION FIVE OR MORE Amine Asselah We show a remarkable similarity between strategies to realize a large intersection or self-intersection local times in dimension five or more. This leads to the same rate functional for large deviation principles for the two objects obtained respectively by Chen and Morters, and by the present author. We also present a new estimate for the distribution of high level sets for a random walk, with application to the geometry of the intersection set of two high level sets of the local times of two independent random walks. http://arxiv.org/abs/0801.3918 --------------------------------------------------------------- 6615. ON THE INFIMUM CONVOLUTION INEQUALITY Rafa{\l} Lata{\l}a and Jakub Onufry Wojtaszczyk In the paper we study the infimum convolution inequalites. Such an inequality was first introduced by B. Maurey to give the optimal concentration of measure behaviour for the product exponential measure. We show how IC- inequalities are tied to concentration and study the optimal cost functions for an arbitrary probability measure. In particular, we show the optimal IC-inequality for product log-concave measures and for uniform measures on the l_p^n balls. Such an optimal inequality implies, for a given measure, in particular the Central Limit Theorem of Klartag and the tail estimates of Paouris. http://arxiv.org/abs/0801.4036 --------------------------------------------------------------- 6616. CARRY PROPAGATION IN MULTIPLICATION BY CONSTANTS Alexander Izsak and Nicholas Pippenger Suppose that a random n-bit number V is multiplied by an odd constant M, greater than or equal to 3, by adding shifted versions of the number V corresponding to the 1s in the binary representation of the constant M. Suppose further that the additions are performed by carry-save adders until the number of summands is reduced to two, at which time the final addition is performed by a carry-propagate adder. We show that in this situation the distribution of the length of the longest carry-propagation chain in the final addition is the same (up to terms tending to 0 as n tends to infinity) as when two independent n-bit numbers are added, and in particular the mean and variance are the same (again up to terms tending to 0). This result applies to all possible orders of performing the carry-save additions. http://arxiv.org/abs/0801.4040 --------------------------------------------------------------- 6617. NO ARBITRAGE CONDITIONS FOR SIMPLE TRADING STRATEGIES Erhan Bayraktar and Hasanjan Sayit Strict local martingales may admit arbitrage opportunities with respect to the class of simple trading strategies. (Since there is no possibility of using doubling strategies in this framework, the losses are not assumed to be bounded from below.) We show that for a class of non-negative strict local martingales, the strong Markov property implies the no arbitrage property with respect to the class of simple trading strategies. This result can be seen as a generalization of a similar result on three dimensional Bessel process in [3]. We also pro- vide no arbitrage conditions for stochastic processes within the class of simple trading strategies with shortsale restriction. http://arxiv.org/abs/0801.4047 --------------------------------------------------------------- 6618. CRITICAL PERCOLATION ON CAYLEY GRAPHS OF GROUPS ACTING ON TREES Iva Kozakova This article presents a method for finding the critical probability $p_c$ for the Bernoulli bond percolation on graphs with the so called tree-like structure. Such graphs can be decomposed into a tree of pieces which have finitely many isomorphism classes. This class of graphs includes the Cayley graphs of amalgamated products, HNN extensions or general groups acting on trees. It also includes all transitive graphs with more than one end. The idea of the method is to find a multi-type Galton-Watson branching process (with a parameter $p$) which has finite expected population size if and only if the expected percolation cluster size is finite. This provides a sufficient information about $p_c$. In particular if the pairwise intersections of pieces are finite, then $p_c$ is the smallest positive $p$ for which $\det(M-1)=0$, where $M$ is the first-moment matrix of the branching process. If the pieces of the tree-like structure are finite, then $p_c$ is an algebraic number, and we give an algorithm computing $p_c$ as a root of some algebraic function. We show that any Cayley graph of a group acting on a tree with finite vertex stabilizers with respect to any finite generating set has a tree-like structure with finite pieces. In particular we show how to compute $p_c$ of the Cayley graph of a free group with respect to any finite generating set. http://arxiv.org/abs/0801.4153 --------------------------------------------------------------- 6619. FROM COMBINATORICS TO LARGE DEVIATIONS FOR THE INVARIANT MEASURES OF SOME MULTICLASS PARTICLE SYSTEMS Davide Gabrielli We prove large deviation principles (LDP) for the invariant measures of the multiclass totally asymmetric simple exclusion process (TASEP) and the multiclass Hammersely-Aldous-Diaconis (HAD) process on a torus. The proof is based on a combinatorial representation of the measures in terms of a \emph{collapsing procedure} introduced in \cite{A} for the 2-class TASEP and then generalized in \cite{FM1}, \cite{FM2} and \cite{FM3} to the multiclass TASEP and the multiclass HAD process. The rate functionals are written in terms of variational problems that we solve in the cases of 2-class processes. http://arxiv.org/abs/0801.4156 --------------------------------------------------------------- 6620. FORECASTING VOLATILITY WITH THE MULTIFRACTAL RANDOM WALK MODEL Jean Duchon (IF) and Raoul Robert (IF) and Vincent Vargas (CEREMADE) We study the problem of forecasting volatility for the multifractal random walk model. In order to avoid the ill posed problem of estimating the correlation length T of the model, we introduce a limiting object defined in a quotient space; formally, this object is an infinite range logvolatility. For this object and the non limiting object, we obtain precise prediction formulas and we apply them to the problem of forecasting volatility and pricing options with the MRW model in the absence of a reliable estimate of the average volatility and T. http://arxiv.org/abs/0801.4220 --------------------------------------------------------------- 6621. A PERMUTATION MODEL FOR FREE RANDOM VARIABLES AND ITS CLASSICAL ANALOGUE Florent Benaych-Georges (PMA) and Ion Nechita (ICJ) In this paper, we generalize a permutation model for free random variables which was first proposed by Biane in \cite{biane}. We also construct its classical probability analogue, by replacing the group of permutations with the group of subsets of a finite set endowed with the symmetric difference operation. These models provide explicit examples of non random matrices which are asymptotically free or independent. The moments and the free (resp. classical) cumulants of the limiting distributions are expressed in terms of a special subset of (noncrossing) pairings. At the end of the paper we present some combinatorial applications of our results. http://arxiv.org/abs/0801.4229 --------------------------------------------------------------- 6622. LOWER BOUNDS FOR TRANSITION PROBABILITIES ON GRAPHS Andras Telcs The paper presents two results. The first one provides separate conditions for the upper and lower estimate of the distribution of the exit time from balls of a random walk on a weighted graph. The main result of the paper is that the lower estimate follows from the elliptic Harnack inequality. The second result is an off-diagonal lower bound for the transition probability of the random walk. http://arxiv.org/abs/0801.4260 --------------------------------------------------------------- 6623. NECESSARY AND SUFFICIENT OPTIMALITY CONDITIONS FOR RELAXED AND STRICT CONTROL PROBLEMS Seid Bahlali We consider a stochastic control problem where the set of strict (classical) controls is not necessarily convex, and the system is governed by a nonlinear stochastic differential equation, in which the control enters both the drift and the diffusion coefficients. By introducing a new approach, we establish necessary as well as sufficient conditions of optimality for two models. The first concerns the relaxed controls, who are a measure-valued processes in which an optimal solution exists. The second is a particular case of the first and relates to strict control problems. These results are given in the form of global stochastic maximum principle by using only the first order expansion and the associated adjoint equation. This improves all the previous works on the subject. http://arxiv.org/abs/0801.4285 --------------------------------------------------------------- 6624. NECESSARY AND SUFFICIENT OPTIMALITY CONDITIONS FOR RELAXED AND STRICT CONTROL PROBLEMS OF FORWARD-BACKWARD SYSTEMS Seid Bahlali We consider a stochastic control problem of nonlinear forward-backward systems, where the set of strict (classical) controls need not be convex and the coefficients depend explicitly on the variable control. By introducing a new approach, we establish necessary as well as sufficient conditions of optimality, in the form of global stochastic maximum principle, for two models. The first concerns the relaxed controls, who are a measure-valued processes. The second is a restriction of the first to strict control problems http://arxiv.org/abs/0801.4326 --------------------------------------------------------------- 6625. THE STABILITY OF CONDITIONAL MARKOV PROCESSES AND MARKOV CHAINS IN RANDOM ENVIRONMENTS Ramon van Handel We consider a discrete time hidden Markov model where the signal is a stationary Markov chain. When conditioned on the observations, the signal is a Markov chain with stationary transition probabilities under the conditional measure. It is shown that this conditional signal is weakly ergodic when the signal is weakly ergodic and the observations are nondegenerate. This permits a delicate exchange of the intersection and supremum of sigma-fields, which has direct implications for the stability of nonlinear filters. The proof relies on an extension of results on the weak ergodicity of Markov chains in random environments to general state spaces. Finally it is shown that the main results can be lifted to the continuous time setting. The results partially resolve a long-standing gap in the proof of a result of H. Kunita (1971). http://arxiv.org/abs/0801.4366 --------------------------------------------------------------- 6626. ASYMPTOTICS FOR THE SURVIVAL PROBABILITY OF A ROUSE CHAIN MONOMER G.Oshanin (LPTMC and University of Paris 6 and Paris and France) We study the long-time asymptotical behavior of the survival probability P_t of a tagged monomer of an infinitely long Rouse chain in presence of two fixed absorbing boundaries, placed at x = \pm L. Mean-square displacement of a tagged monomer obeys \bar{X^2(t)} \sim t^{1/2} at all times, which signifies that its dynamics is an anomalous diffusion process. Constructing lower and upper bounds on P_t, which have the same time-dependence but slightly differ by numerical factors in the definition of the characteristic relaxation time, we show that P_t is a stretched- exponential function of time, \ln(P_t) \sim - t^{1/2}/L^2. This implies that the distribution function of the first exit time from a fixed interval [- L,L] for such an anomalous diffusion has all moments. http://arxiv.org/abs/0801.2914 --------------------------------------------------------------- 6627. FIRST-EXIT-TIME PROBABILITY DENSITY TAILS FOR A LOCAL HEIGHT OF A NON-EQUILIBRIUM GAUSSIAN INTERFACE G.Oshanin (LPTMC and University of Paris 6 and France) We study the long-time behavior of the probability density Q_t of the first exit time from a bounded interval [-L,L] for a stochastic non- Markovian process h(t) describing fluctuations at a given point of a two-dimensional, infinite in both directions Gaussian interface. We show that Q_t decays when t \to \infty as a power-law $^{-1 - \alpha}, where \alpha is non-universal and proportional to the ratio of the thermal energy and the elastic energy of a fluctuation of size L. The fact that \alpha appears to be dependent on L, which is rather unusual, implies that the number of existing moments of Q_t depends on the size of the window [-L,L]. A moment of an arbitrary order n, as a function of L, exists for sufficiently small L, diverges when L approaches a certain threshold value L_n, and does not exist for L > L_n. For L > L_1, the probability density Q_t is normalizable but does not have moments. http://arxiv.org/abs/0801.3975 --------------------------------------------------------------- 6628. CANONICAL MOMENTS AND RANDOM SPECTRAL MEASURES Fabrice Gamboa Alain Rouault We study some connections between the random moment problem and the random matrix theory. A uniform pick in a space of moments can be lifted into the spectral probability measure of the pair (A;e) where A is a random matrix from a classical ensemble and e is a fixed unit vec- tor. This random measure is a weighted sampling among the eigenvalues of A. We also study the large deviations properties of this random measure when the dimension of the matrix grows. The rate function for these large deviations involves the reversed Kullback information. http://arxiv.org/abs/0801.4400 --------------------------------------------------------------- 6629. ON THE BIRTH-AND-ASSASSINATION PROCESS, WITH AN APPLICATION TO SCOTCHING A RUMOR IN A NETWORK Charles Bordenave We give new formulas on the total number of born particles in the stable birth-and-assassination process, and prove that it has an heavy-tailed distribution. We also establish that this process is a scaling limit of a process of rumor scotching in a network, and is related to a predator- prey dynamics. http://arxiv.org/abs/0801.4499 --------------------------------------------------------------- 6630. SUBORDINATED DISCRETE SEMIGROUPS OF OPERATORS Nick Dungey Given a power-bounded linear operator T in a Banach space and a probability F on the non-negative integers, one can form a `subordinated' operator S = \sum_k F(k) T^k. We obtain asymptotic properties of the subordinated discrete semigroup (S^n: n=1,2,...) under certain conditions on F. In particular, we study probabilities F with the property that S satisfies the Ritt resolvent condition whenever T is power-bounded. Examples and counterexamples of this property are discussed. The hypothesis of power-boundedness of T can sometimes be replaced by the weaker Kreiss resolvent condition. http://arxiv.org/abs/0801.4557 --------------------------------------------------------------- 6631. CONVEX ORDERING FOR RANDOM VECTORS USING PREDICTABLE REPRESENTATION Marc Arnaudon (LMA) and Jean-Christophe Breton (LMCA) and Nicolas Privault We prove convex ordering results for random vectors admitting a predictable representation in terms of a Brownian motion and a non-necessarily independent jump component. Our method uses forward-backward stochastic calculus and extends previous results in the one-dimensional case. We also study a geometric interpretation of convex ordering for discrete measures in connection with the conditions set on the jump heights and intensities of the considered processes. http://arxiv.org/abs/0801.4621 --------------------------------------------------------------- 6632. REFRACTED LEVY PROCESSES AND RUIN Andreas E. Kyprianou and Ronnie Loeffen Motivated by classical considerations from the theory of risk theory we investigate the problem of ruin for a so-called refracted L\'evy process. The latter is a L\'evy processes whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More formally, whenever it exists, a refracted L \'evy process is described by the unique weak solution to the stochastic differential equation \[ \D U_t = - \delta \mathbf{1}_{(U_t >b)}\D t + \D X_t \] where $X=\{X_t :t\geq 0\}$ is a L\'evy process with law $\mathbb{P}$ and $b, \delta\in \mathbb{R}$ such that the resulting process $U$ may visit the half line $(b,\infty)$ with positive probability. In the light of connection with a certain dividend payment strategy on risk processes, we are particularly interested in the case that $X$ is spectrally negative, $b>0$ and $0< \delta<\mathbb{E}(X_1)$. For that case we provide some new identities for certain functionals of the path of the refracted process which are of relevance to the ruin problem. http://arxiv.org/abs/0801.4655 --------------------------------------------------------------- 6633. STOCHASTIC MAXIMUM PRINCIPLE FOR OPTIMAL CONTROL PROBLEM OF BACKWARD SYSTEMS WITH TERMINAL CONDITION IN L1 Seid Bahlali We consider a stochastic control problem, where the control domain is convex and the system is governed by a nonlinear backward stochastic differential equation. With a L1 terminal data, we derive necessary optimality conditions in the form of stochastic maximum principle. http://arxiv.org/abs/0801.4666 --------------------------------------------------------------- 6634. THE STRICT AND RELAXED STOCHASTIC MAXIMUM PRINCIPLE FOR OPTIMAL CONTROL PROBLEM OF BACKWARD SYSTEMS Seid Bahlali We consider a stochastic control problem where the set of controls is not necessarily convex and the system is governed by a nonlinear backward stochastic differential equation. We establish necessary as well as sufficient conditions of optimality for two models. The first concerns the strict (classical) controls. The second is an extension of the first to relaxed controls, who are a measure valued processes. http://arxiv.org/abs/0801.4668 --------------------------------------------------------------- 6635. A GENERAL STOCHASTIC MAXIMUM PRINCIPLE FOR MIXED RELAXED- SINGULAR CONTROL PROBLEMS Seid Bahlali We consider in this paper, mixed relaxed-singular stochastic control problems, where the control variable has two components, the first being measure-valued and the second singular. The control domain is not necessarily convex and the system is governed by a nonlinear stochastic differential equation, in which the measure-valued part of the control enters both the drift and the diffusion coefficients. We establish necessary optimality conditions, of the Pontryagin maximum principle type, satisfied by an optimal relaxed-singular control, which exist under general conditions on the coefficients. The proof is based on the strict singular stochastic maximum principle established by Bahlali-Mezerdi, Ekeland's variational principle and some stability properties of the trajectories and adjoint processes with respect to the control variable. http://arxiv.org/abs/0801.4669 --------------------------------------------------------------- 6636. GRADIENT ESTIMATE AND HARNACK INEQUALITY ON NON-COMPACT RIEMANNIAN MANIFOLDS Marc Arnaudon (LMA) and Anton Thalmaier and Feng-Yu Wang A new type of gradient estimate is established for diffusion semigroups on non-compact complete Riemannian manifolds. As applications, a global Harnack inequality with power and a heat kernel estimate are derived for diffusion semigroups on arbitrary complete Riemannian manifolds. http://arxiv.org/abs/0801.4708 --------------------------------------------------------------- 6637. THE BERNOULLI SIEVE REVISITED Alexander Gnedin and Alex Iksanov and Pavlo Negadajlov and Uwe Roesler We consider an occupancy scheme in which `balls' are identified with $n$ points sampled from the standard exponential distribution, while the role of `boxes' is played by the spacings induced by an independent random walk with positive and non-lattice steps. We discuss the asymptotic behaviour of five quantities: the index $K_n^*$ of the last occupied box, the number $K_n $ of occupied boxes, the number $K_{n,0}$ of empty boxes whose index is at most $K_n^*$, the index $W_n$ of the first empty box and the number of balls $Z_n$ in the last occupied box. It is shown that the limiting distribution of properly scaled and centered $K_n^*$ coincides with that of the number of renewals not exceeding $\log n$. A similar result is shown for $K_n$ and $W_n$ under a side condition that prevents occurrence of very small boxes. The condition also ensures that $K_{n,0}$ converges in distribution. Limiting results for $Z_n$ are established under an assumption of regular variation. http://arxiv.org/abs/0801.4725 --------------------------------------------------------------- 6638. STOCHASTIC EXTREMA AS STATIONARY PHASES OF CHARACTERISTIC FUNCTIONS S. Nikitin The paper is dealing with semi-classical asymptotics of a characteristic function for a stochastic process. The main technical tool is provided by the stationary phase method. The extremal range for a stochastic process is defined by limit values of the complex logarithm of the characteristic function. The paper also outlines a numerical method for calculating stochastic extrema. http://arxiv.org/abs/0801.4726 --------------------------------------------------------------- 6639. HEDGING STRATEGIES AND MINIMAL VARIANCE PORTFOLIOS FOR EUROPEAN AND EXOTIC OPTIONS IN A LEVY MARKET Wing Yan Yip and Sofia Olhede and David Stephens This paper presents hedging strategies for European and exotic options in a Levy market. By applying Taylor's Theorem, dynamic hedging portfolios are con- structed under different market assumptions, such as the existence of power jump assets or moment swaps. In the case of European options or baskets of European options, static hedging is implemented. It is shown that perfect hedging can be achieved. Delta and gamma hedging strategies are extended to higher moment hedging by investing in other traded derivatives depending on the same underlying asset. This development is of practical importance as such other derivatives might be readily available. Moment swaps or power jump assets are not typically liquidly traded. It is shown how minimal variance portfolios can be used to hedge the higher order terms in a Taylor expansion of the pricing function, investing only in a risk-free bank account, the underlying asset and potentially variance swaps. The numerical algorithms and performance of the hedging strategies are presented, showing the practical utility of the derived results. http://arxiv.org/abs/0801.4941 --------------------------------------------------------------- 6640. STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTION AND STANDARD BROWNIAN MOTION Jo\~ao Guerra and David Nualart We prove an existence and uniqueness theorem for solutions of multidimensional, time dependent, stochastic differential equations driven simultaneously by a multidimensional fractional Brownian motion with Hurst parameter H>1/2 and a multidimensional standard Brownian motion. The proof relies on some a priori estimates, which are obtained using the methods of fractional integration, and the classical Ito stochastic calculus. The existence result is based on the Yamada-Watanabe theorem. http://arxiv.org/abs/0801.4963 --------------------------------------------------------------- 6641. DEGENERATE STOCHASTIC DIFFERENTIAL EQUATIONS FOR CATALYTIC BRANCHING NETWORKS Sandra M. Kliem Uniqueness of the martingale problem corresponding to a degenerate SDE which models catalytic branching networks is proven. This work is an extension of a paper by Dawson and Perkins to arbitrary catalytic branching networks. As part of the proof estimates on the corresponding semigroup are found in terms of weighted Holder norms for arbitrary networks, which are proven to be equivalent to the semigroup norm for this generalized setting. ----- On prouve l'unicite d'un probleme de martingale correspondant a une EDS degeneree, qui apparait comme un modele de reseaux avec branchement catalytique. Ce travail est une extension des resultats de Dawson et Perkins au cas de reseaux generaux. On obtient en particulier des estimees pour le semi-groupe des reseaux generaux, sous forme de normes de Holder ponderees; et on etablit l'equivalence de ces normes avec des normes de semi-groupe dans ce contexte general. http://arxiv.org/abs/0802.0035 --------------------------------------------------------------- 6642. ON THE SUPREMUM OF RANDOM DIRICHLET POLYNOMIALS WITH MULTIPLICATIVE COEFFICIENTS Mikhail Lifshits and Michel Weber We study the supremum of some random Dirichlet polynomials with independent coefficients and obtain sharp upper and lower bounds for supremum expectation thus extending the results from our previous work (see http://arXiv.org/abs/math/0703691). Our approach in proving these results is entirely based on methods of stochastic processes, in particular the metric entropy method. http://arxiv.org/abs/0802.0071 --------------------------------------------------------------- 6643. LIMIT THEOREMS FOR LARGE DIMENSIONAL SAMPLE MEANS, SAMPLE COVARIANCE MATRICES AND HOTELLING'S T^2 STATISTICS Guangming Pan and Wang Zhou In this paper, we prove the central limit theorem for Hotelling's $T^2$ statistics when the dimension of the random vectors is proportional to the sample size via investigating asymptotic independence and random quadratic forms involving sample means and sample covariance matrices. http://arxiv.org/abs/0802.0082 --------------------------------------------------------------- 6644. OCCUPATION TIME FLUCTUATION LIMITS OF INFINITE VARIANCE EQUILIBRIUM BRANCHING SYSTEMS Piotr Milos We establish limit theorems for the fluctuations of the rescaled occupation time of a $(d,\alpha,\beta)$-branching particle system. It consists of particles moving according to a symmetric $\alpha$-stable motion in $\mathbb{R}^d$. The branching law is in the domain of attraction of a (1+$\beta$)-stable law and the initial condition is an equilibrium random measure for the system (defined below). In the paper we treat separately the cases of intermediate $\alpha/\beta(1+\beta)\alpha/\beta $ dimensions. In the most interesting case of intermediate dimensions we obtain a version of a fractional stable motion. The long-range dependence structure of this process is also studied. Contrary to this case, limit processes in critical and large dimensions have independent increments. http://arxiv.org/abs/0802.0187 --------------------------------------------------------------- 6645. V-VARIABLE FRACTALS: FRACTALS WITH PARTIAL SELF SIMILARITY Michael Barnsley and John E. Hutchinson and \"Orjan Stenflo We establish properties of a new type of fractal which has partial self similarity at all scales. For any collection of iterated functions systems with an associated probability distribution and any positive integer V there is a corresponding class of V-variable fractal sets or measures. These V- variable fractals can be obtained from the points on the attractor of a single deterministic iterated function system. Existence, uniqueness and approximation results are established under average contractive assumptions. We also obtain extensions of some basic results concerning iterated function systems. http://arxiv.org/abs/0802.0064 --------------------------------------------------------------- 6646. HARNACK INEQUALITY AND APPLICATIONS FOR STOCHASTIC EVOLUTION EQUATIONS WITH MONOTONE DRIFT Wei Liu In this paper, the dimension-free Harnack inequality is proved for transition semigroups of solutions to a large class of stochastic evolution equations with monotone drift. As a conseqence, the strong Feller property, ergodic property and hyper-(or ultra-)contractivity are established for corresponding semigroups. The main results can be applied to many concrete stochastic evolution equations such as stochastic reaction-diffusion equation, stochastic p-Laplacian equation in Hilbert space. http://arxiv.org/abs/0802.0289 --------------------------------------------------------------- 6647. NONDIFFERENTIABLE FUNCTIONS OF ONE DIMENSIONAL SEMIMARTINGALES George Lowther In this paper we consider decompositions of processes of the form Y=f(t,X) where X is a one dimensional semimartingale, but f is not required to be differentiable so Ito's formula does not apply. First, in the case where f(t,x) is independent of t, we show that requiring it to be locally Lipschitz continuous in x is enough for an Ito style decomposition to apply. This decomposes Y into a stochastic integral term and a term whose quadratic variation is well defined and has zero continuous part. For the time dependent case we show that the same decomposition still holds under the additional conditions that the left and right derivatives of f(t,x) in x exist, it is right-continuous in t, and that locally its variation with respect to t is integrable in x. In particular, in the continuous case this shows that Y is a Dirichlet process. We furthermore prove that such processes satisfy a decomposition into continuous martingale and purely discontinuous terms, and a Doob-Meyer style decomposition. http://arxiv.org/abs/0802.0331 --------------------------------------------------------------- 6648. IMPROVED MIXING TIME BOUNDS FOR THE THORP SHUFFLE AND L-REVERSAL CHAIN Ben Morris We prove a theorem that reduces bounding the mixing time of a card shuffle to verifying a condition that involves only pairs of cards, then we use it to obtain improved bounds for two previously studied models. E. Thorp introduced the following card shuffling model in 1973. Suppose the number of cards n is even. Cut the deck into two equal piles. Drop the first card from the left pile or from the right pile according to the outcome of a fair coin flip. Then drop from the other pile. Continue this way until both piles are empty. We obtain a mixing time bound of O(log^4 n). Previously, the best known bound was O(log^{29} n) and previous proofs were only valid for n a power of 2. We also analyze the following model, called the L-reversal chain, introduced by Durrett. There are n cards arrayed in a circle. Each step, an interval of cards of length at most L is chosen uniformly at random and its order is reversed. Durrett has conjectured that the mixing time is O(max(n, n^3/ L^3) log n). We obtain a bound that is within a factor O(log^2 n) of this,the first bound within a poly log factor of the conjecture. http://arxiv.org/abs/0802.0339 --------------------------------------------------------------- 6649. EXACT EXPONENTIAL BOUNDS FOR THE RANDOM FIELD MAXIMUM DISTRIBUTION VIA THE MAJORING MEASURES (GENERIC CHAINING) E. Ostrovsky and E. Rogover In this paper non-asymptotic exact exponential estimates are derived for the tail of maximum distribution of random field in the terms of majoring measures or, equally, generic chaining. http://arxiv.org/abs/0802.0349 --------------------------------------------------------------- 6650. JENSEN'S INEQUALITY FOR G-CONVEX FUNCTION UNDER G-EXPECTATION Guangyan Jia and Shige Peng A real valued function defined on}$\mathbb{R}$ {\small is called}$g$ {\small --convex if it satisfies the following \textquotedblleft generalized Jensen's inequality\textquotedblright under a given}$g${\small -expectation, i.e., }$h(\mathbb{E}^{g}[X])\leq \mathbb{E}% ^{g}[h(X)]${\small, for all random variables}$X$ {\small such that both sides of the inequality are meaningful. In this paper we will give a necessary and sufficient conditions for a }$C^{2}${\small -function being}$% g ${\small -convex. We also studied some more general situations. We also studied}$g${\small -concave and}$g$ {\small -affine functions. http://arxiv.org/abs/0802.0373 --------------------------------------------------------------- 6651. ABSOLUTE CONTINUITY AND SINGULARITY OF TWO PROBABILITY MEASURES ON A FILTERED SPACE S.S. Gabriyelyan Let $\mu$ and $\nu$ be fixed probability measures on a filtered space $(\Omega, ({\cal F}_t)_{t\in {\bf R}^{+}}, {\cal F})$. Denote by $ \mu_T $ and $\nu_T $ (respectively, $\mu_{T-} $ and $\nu_{T-} $) the restrictions of measures $\mu$ and $\nu$ on ${\cal F}_T $ (respectively, on ${\cal F}_{T-} $) for a stopping time $T$. We can find a Hahn-decomposition of $\mu_T $ and $\nu_T $ using a Hahn-decomposition of measures $\mu$, $\nu$, and a Hellinger process $h_t$ in the strict sense of order 1/2. The norm of the absolutely continuity component of $\mu_{T-} $ relative to $\nu_{T-} $ in terms of density processes and Hellinger integrals is computed. http://arxiv.org/abs/0802.0385 --------------------------------------------------------------- 6652. A QUADRATIC REGRESSION PROBLEM FOR TWO-STATE ALGEBRAS WITH APPLICATION TO THE CENTRAL LIMIT THEOREM Marek Bozejko and Wlodzimierz Bryc We extend a free version of the Laha-Lukacs theorem to probability spaces with two-states. We then use this result to generalize a noncommutative CLT of Kargin to the two-state setting. http://arxiv.org/abs/0802.0266 --------------------------------------------------------------- 6653. JACK POLYNOMIALS AND FREE CUMULANTS Michel Lassalle (CNRS and Marne la Vallee and France) We study the coefficients in the expansion of Jack polynomials in terms of power sums. We express them as polynomials in the free cumulants of the transition measure of an anisotropic Young diagram. We conjecture that such polynomials have nonnegative integer coefficients. This extends recent results about normalized characters of the symmetric group. http://arxiv.org/abs/0802.0448 --------------------------------------------------------------- 6654. LARGE DEVIATIONS FOR THE STOCHASTIC SHELL MODEL OF TURBULENCE U. Manna and S.S. Sritharan and and P. Sundar In this work we first prove the existence and uniqueness of a strong solution to stochastic GOY model of turbulence with a small multiplicative noise. Then using the weak convergence approach, Laplace principle for so- lutions of the stochastic GOY model is established in certain Polish space. Thus a Wentzell-Freidlin type large deviation principle is established utilizing certain results by Varadhan and Bryc. http://arxiv.org/abs/0802.0585 --------------------------------------------------------------- 6655. A UNIQUENESS THEOREM FOR SOLUTION OF BSDES Guangyan Jia In this note, we prove that if $g$ is uniformly continuous in $z$, uniformly with respect to $(\oo,t)$ and independent of $y$, the solution to the backward stochastic differential equation (BSDE) with generator $g$ is unique. http://arxiv.org/abs/0802.0616 --------------------------------------------------------------- 6656. MULTIFRACTIONAL, MULTISTABLE, AND OTHER PROCESSES WITH PRESCRIBED LOCAL FORM K.J. Falconer and J. Levy Vehel We present a general method for constructing stochastic processes with prescribed local form. Such processes include variable amplitude multifractional Brownian motion, multifractional $\alpha$-stable processes, and multistable processes, that is processes that are locally $\alpha(t)$- stable but where the stability index $\alpha(t)$ varies with $t$. In particular we construct multifractional multistable processes where both the local self-similarity and stability indices vary. http://arxiv.org/abs/0802.0645 --------------------------------------------------------------- 6657. STABILIZATION AND LIMIT THEOREMS FOR GEOMETRIC FUNCTIONALS OF GIBBS POINT PROCESSES T. Schreiber and J. E. Yukich Given a Gibbs point process $\P^{\Psi}$ on $\R^d$ having a weak enough potential $\Psi$, we consider the random measures $\mu_\la := \sum_{x \in \P^{\Psi} \cap Q_\la} \xi(x, \P^{\Psi} \cap Q_\la) \delta_{x/\la^{1/d}} $, where $Q_{\la} := [-\la^{1/d}/2,\la^{1/d}/2]^d$ is the volume $\la$ cube and where $\xi(\cdot,\cdot)$ is a translation invariant stabilizing functional. Subject to $\Psi$ satisfying a localization property and translation invariance, we establish weak laws of large numbers for $\la^{-1} \mu_\la(f)$, $f$ a bounded test function on $\R^d$, and weak convergence of $\la^{-1/2} \mu_ \la(f),$ suitably centered, to a Gaussian field acting on bounded test functions. The result yields limit laws for geometric functionals on Gibbs point processes including the Strauss and area interaction point processes as well as more general point processes defined by the Widom-Rowlinson and hard-core model. We provide applications to random sequential packing on Gibbsian input, to functionals of Euclidean graphs, networks, and percolation models on Gibbsian input, and to quantization via Gibbsian input. http://arxiv.org/abs/0802.0647 --------------------------------------------------------------- 6658. FRACTIONAL CAUCHY PROBLEMS ON BOUNDED DOMAINS Mark M. Meerschaert and Erkan Nane and Palaniappan Vellaisamy Fractional Cauchy problems replace the usual first order time derivative by a fractional derivative. This paper develops classical solutions and stochastic analogues for fractional Cauchy problems in a bounded domain $D\subset \rd$ with Dirichlet boundary conditions. Stochastic solutions are constructed via an inverse stable subordinator whose scaling index corresponds to the order of the fractional time derivative. Dirichlet problems corresponding to iterated Brownian motion in a bounded domain are then solved by establishing a correspondence with the case of a half-derivative in time. http://arxiv.org/abs/0802.0673 --------------------------------------------------------------- 6659. ON THE ASYMPTOTIC NORMALITY OF THE CONDITIONAL MAXIMUM LIKELIHOOD ESTIMATORS FOR THE TRUNCATED REGRESSION MODEL AND THE TOBIT MODEL Chunlin Wang In this paper, we study the asymptotic normality of the conditional maximum likelihood (ML) estimators for the truncated regression model and the Tobit model. We show that under the general setting assumed in his book, the conjectures made by Hayashi (2000) \footnote{see page 516, and page 520 of Hayashi (2000).} about the asymptotic normality of the conditional ML estimators for both models are true, namely, a sufficient condition is the nonsingularity of $\mathbf{x_tx'_t}$. http://arxiv.org/abs/0802.0536 --------------------------------------------------------------- 6660. ON THE DISTRIBUTION OF THE DOMINATION NUMBER OF A NEW FAMILY OF PARAMETRIZED RANDOM DIGRAPHS E. Ceyhan and C. E. Priebe We derive the asymptotic distribution of the domination number of a new family of random digraph called proximity catch digraph (PCD), which has application to statistical testing of spatial point patterns and to pattern recognition. The PCD we use is a parametrized digraph based on two sets of points on the plane, where sample size and locations of the elements of one is held fixed, while the sample size of the other whose elements are randomly distributed over a region of interest goes to infinity. PCDs are constructed based on the relative allocation of the random set of points with respect to the Delaunay triangulation of the other set whose size and locations are fixed. We introduce various auxiliary tools and concepts for the derivation of the asymptotic distribution. We investigate these concepts in one Delaunay triangle on the plane, and then extend them to the multiple triangle case. The methods are illustrated for planar data, but are applicable in higher dimensions also. http://arxiv.org/abs/0802.0617 --------------------------------------------------------------- 6661. TIME--SPACE WHITE NOISE ELIMINATES GLOBAL SOLUTIONS IN REACTION DIFFUSION EQUATIONS Juli\'an Fern\'andez Bonder and Pablo Groisman We prove that perturbing the reaction--diffusion equation $u_t=u_{xx} + (u_+)^p$ ($p>1$), with time--space white noise produces that solutions explodes with probability one for every initial datum, opposite to the deterministic model where a positive stationary solution exists. http://arxiv.org/abs/0802.0633 --------------------------------------------------------------- 6662. SEMICLASSICAL ANALYSIS OF A RANDOM WALK ON A MANIFOLD G. Lebeau and L. Michel We prove sharp rate of convergence to stationarity for a natural random walk on a compact Riemannian manifold (M,g). The proof includes a detailed study of the spectral theory of the associated operator. http://arxiv.org/abs/0802.0644 --------------------------------------------------------------- 6663. ON THE LOCAL TIME OF THE ASYMMETRIC BERNOULLI WALK Endre Cs\'aki and Ant\'onia F\"oldes and P\'al R\'ev\'esz We study some properties of the local time of the asymmetric Bernoulli walk on the line. These properties are very similar to the corresponding ones of the simple symmetric random walks in higher ($d\geq3$) dimension, which we established in the recent years. The goal of this paper is to highlight these similarities. http://arxiv.org/abs/0802.0765 --------------------------------------------------------------- 6664. TRANSIENT NEAREST NEIGHBOR RANDOM WALK AND BESSEL PROCESS Endre Cs\'aki and Ant\'onia F\"oldes and P\'al R\'ev\'esz We prove strong invariance principle between a transient Bessel process and a certain nearest neighbor (NN) random walk that is constructed from the former by using stopping times. It is also shown that their local times are close enough to share the same strong limit theorems. It is shown furthermore, that if the difference between the distributions of two NN random walks are small, then the walks themselves can be constructed so that they are close enough. Finally, some consequences concerning strong limit theorems are discussed. http://arxiv.org/abs/0802.0778 --------------------------------------------------------------- 6665. ON THE LAMPERTI STABLE PROCESSES M.E. Caballero and J.C. Pardo and J.L. P\'erez We consider a new family of $\R^d$-valued L\'{e}vy processes that we call Lamperti stable. One of the advantages of this class is that the law of many related functionals can be computed explicitely (see for instance \cite{cc}, \cite{ckp}, \cite{kp} and \cite{pp}). This family of processes shares many properties with the tempered stable and the layered stable processes, defined in Rosi\'nski \cite{ro} and Houdr\'e and Kawai \cite{hok} respectively, for instance their short and long time behaviour. Additionally, in the real valued case we find a series representation which is used for sample paths simulation. In this work we find general properties of this class and we also provide many examples, some of which appear in recent literature. http://arxiv.org/abs/0802.0851 --------------------------------------------------------------- 6666. CONTINUOUS LOCAL TIME OF A PURELY ATOMIC IMMIGRATION SUPERPROCESS WITH DEPENDENT SPATIAL MOTION Zenghu Li and Jie Xiong A purely atomic immigration superprocess with dependent spatial motion in the space of tempered measures is constructed as the unique strong solution of a stochastic integral equation driven by Poisson processes based on the excursion law of a Feller branching diffusion, which generalizes the work of Dawson and Li (2003). As an application of the stochastic equation, it is proved that the superprocess possesses a local time which is Holder continuous of order $\alpha$ for every $\alpha< 1/2$. We establish two scaling limit theorems for the immigration superprocess, from which we derive scaling limits for the corresponding local time. http://arxiv.org/abs/0802.0926 --------------------------------------------------------------- 6667. STOCHASTIC EQUATIONS OF NON-NEGATIVE PROCESSES WITH JUMPS Zongfei Fu and Zenghu Li We study stochastic equations of non-negative processes with jumps. The existence and uniqueness of strong solutions are established under Lipschitz and non-Lipschitz conditions. The comparison property of two solutions are proved under suitable conditions. The results are applied to stochastic equations driven by one-sided Levy processes and those of continuous state branching processes with immigration. http://arxiv.org/abs/0802.0933 --------------------------------------------------------------- 6668. EXISTENCE OF NON-TRIVIAL HARMONIC FUNCTIONS ON CARTAN-HADAMARD MANIFOLDS OF UNBOUNDED CURVATURE Marc Arnaudon (LMA) and Anton Thalmaier and Stefanie Ulsamer The Liouville property of a complete Riemannian manifold (i.e., the question whether there exist non-trivial bounded harmonic functions) attracted a lot of attention. For Cartan-Hadamard manifolds the role of lower curvature bounds is still an open problem. We discuss examples of Cartan-Hadamard manifolds of unbounded curvature where the limiting angle of Brownian motion degenerates to a single point on the sphere at infinity, but where nevertheless the space of bounded harmonic functions is as rich as in the non-degenerate case. To see the full boundary the point at infinity has to be blown up in a non- trivial way. Such examples indicate that the situation concerning the famous conjecture of Greene and Wu about existence of non-trivial bounded harmonic functions on Cartan-Hadamard manifolds is much more complicated than one might have expected. http://arxiv.org/abs/0802.0966 --------------------------------------------------------------- 6669. FINITE SIZE SCALING FOR HOMOGENEOUS PINNING MODELS Julien Sohier (PMA) Pinning models are built from discrete renewal sequences by rewarding (or penalizing) the trajectories according to their number of renewal epochs up to time $N$, and $N$ is then sent to infinity. They are statistical mechanics models to which a lot of attention has been paid both because they are very relevant for applications and because of their {\sl exactly solvable character}, while displaying a non-trivial phase transition (in fact, a localization transition). The order of the transition depends on the tail of the inter-arrival law of the underlying renewal and the transition is continuous when such a tail is sufficiently heavy: this is the case on which we will focus. The main purpose of this work is to give a mathematical treatment of the {\sl finite size scaling limit} of pinning models, namely studying the limit (in law) of the process close to criticality when the system size is proportional to the correlation length. http://arxiv.org/abs/0802.1040 --------------------------------------------------------------- 6670. REPRESENTATION OF THE PENALTY TERM OF DYNAMIC CONCAVE UTILITIES Freddy Delbaen and Shige Peng and Emanuela Rosazza Gianin In this paper we will provide a representation of the penalty term of general dynamic concave utilities (hence of dynamic convex risk measures) by applying the theory of g-expectations. http://arxiv.org/abs/0802.1121 --------------------------------------------------------------- 6671. HIDING THE DRIFT Miklos Rasonyi and Walter Schachermayer and Richard Warnung In this article we consider a Brownian motion with drift, denoted by $S = (S_t)_{t\ge0}$, of the form $dS_t = \mu_t dt + dB_t \qquad \text{for} t \ge 0,$ with a specific non-trivial drift predictable with respect to $ \mathbb{F}^B$, the natural filtration of the Brownian motion $B = (B_t)_{t\ge0}$. We construct a process $H = (H_t)_{t\ge0}$ also predictable with respect to $ \mathbb{F}^B$ such that $((H \cdot S)_t)_{t\ge 0}$ is a Brownian motion in its own filtration. Furthermore, for any $\delta>0$, we refine this construction such that the drift $(\mu_t)_{t\ge0}$ only takes values in $]\mu-\delta,\mu+ \delta[$ for fixed $\mu>0$. http://arxiv.org/abs/0802.1152 --------------------------------------------------------------- 6672. LIMIT THEOREMS FOR HYBRIDIZATION REACTIONS ON OLIGONUCLEOTIDE MICROARRAYS Grzegorz A. Rempala and Iwona Pawlikowska We derive herein the limiting laws for certain stationary distributions of birth-and-death processes related to the classical model of chemical adsorption-desorption reactions due to Langmuir. The model has been recently considered in the context of a hybridization reaction on an oligonucleotide DNA microarray. Our results imply that the truncated gamma- and beta- type distributions can be used as approximations to the observed distributions of the fluorescence readings of the oligo-probes on a microarray. These findings might be useful in developing new model-based, probe-specific methods of extracting target concentrations from array fluorescence readings. http://arxiv.org/abs/0802.1192 --------------------------------------------------------------- 6673. LARGE DEVIATIONS OF LATTICE HAMILTONIAN DYNAMICS COUPLED TO STOCHASTIC THERMOSTATS T. Bodineau and R. Lefevere We discuss the Donsker-Varadhan theory of large deviations in the framework of Hamiltonian systems thermostated by a Gaussian stochastic coupling. We derive a general formula for the Donsker-Varadhan large deviation functional for dynamics which satisfy natural properties under time reversal. Next, we discuss the characterization of the stationary state as the solution of a variational principle and its relation to the minimum entropy production principle. Finally, we compute the large deviation functional of the current in the case of a harmonic chain thermostated by a Gaussian stochastic coupling. http://arxiv.org/abs/0802.1104 --------------------------------------------------------------- 6674. FUNCTION SPACES AND CAPACITY RELATED TO A SUBLINEAR EXPECTATION: APPLICATION TO G-BROWNIAN MOTION PATHES Laurent Denis and Mingshang Hu and Shige Peng In this paper we give some basic and important properties of several typical Banach spaces of functions of $G$-Brownian motion pathes induced by a sublinear expectation--G-expectation. Many results can be also applied to more general situations. A generalized version of Kolmogorov's criterion for continuous modification of a stochastic process is also obtained. http://arxiv.org/abs/0802.1240 --------------------------------------------------------------- 6675. FRACTIONAL TERM STRUCTURE MODELS: NO-ARBITRAGE AND CONSISTENCY Alberto Ohashi In this work we introduce Heath-Jarrow-Morton (HJM) interest rate models driven by fractional Brownian motions. By using support arguments we prove that the resulting model is arbitrage-free under proportional transaction costs in the same spirit of Guasoni et al (2006, 2007). In particular, we obtain a drift condition which is similar in nature to the classical HJM no-arbitrage drift restriction. The second part of this paper deals with consistency problems related to the fractional HJM dynamics. We give a fairly complete characterization of finite-dimensional invariant manifolds for HJM models with fractional Brownian motion by means of Nagumo-type conditions. As an application, we investigate consistency of Nelson-Siegel family with respect to Ho-Lee and Hull- White models. It turns out that similar to the Brownian case such family does not go well with the fractional HJM dynamics with deterministic volatility. In fact, there is no nontrivial fractional interest rate model consistent with the Nelson-Siegel family. http://arxiv.org/abs/0802.1288 --------------------------------------------------------------- 6676. A GENERALIZATION OF DOOB'S MAXIMAL IDENTITY Ashkan Nikeghbali In this paper, using martingale techniques, we prove a generalization of Doob's maximal identity in the setting of continuous nonnegative local submartingales $(X_{t})$ of the form: $X_{t}=N_{t}+A_{t}$, where the measure $(dA_{t})$ is carried by the set $\left\{t: X_{t}=0\right\}$. In particular, we give a multiplicative decomposition for the Az\'ema supermartingale associated with some last passage times related to such processes and we prove that these non-stopping times contain very useful information. As a consequence, we obtain the law of the maximum of a continuous nonnegative local martingale $ (M_t)$ which satisfies $M_\infty=\psi(\sup_{t\geq0}M_t)$ for some measurable function $\psi$ as well as the law of the last time this maximum is reached. http://arxiv.org/abs/0802.1317 --------------------------------------------------------------- 6677. LARGE DEVIATIONS FOR THE BOUSSINESQ EQUATIONS UNDER RANDOM INFLUENCES Jinqiao Duan (IIT) and Annie Millet (CES and Samos and Pma) A Boussinesq model for the Benard convection under random influences is considered as a system of stochastic partial differential equations. This is a coupled system of stochastic Navier-Stokes equations and the transport equation for temperature. Large deviations are proved, using a weak convergence approach based on a variational representation of functionals of infinite dimensional Brownian motion. http://arxiv.org/abs/0802.1335 --------------------------------------------------------------- 6678. ASYMPTOTICS OF THE SPECTRAL GAP FOR THE INTERCHANGE PROCESS ON LARGE HYPERCUBES Shannon Starr and Matt Conomos We consider the interchange process (IP) on the $d$-dimensional, discrete hypercube of side-length $n$. Specifically, we compare the spectral gap of the IP to the spectral gap of the random walk (RW) on the same graph. We prove that the two spectral gaps are asymptotically equivalent, in the limit $n \to \infty$. This result gives further supporting evidence for a conjecture of Aldous, that the spectral gap of the IP equals the spectral gap of the RW on all finite graphs. Our proof is based on an argument invented by Handjani and Jungreis, who proved Aldous's conjecture for all trees. http://arxiv.org/abs/0802.1368 --------------------------------------------------------------- 6679. EXPLICIT COMPUTATIONS FOR A FILTERING PROBLEM WITH POINT PROCESS OBSERVATIONS WITH APPLICATIONS TO CREDIT RISK Vincent Leijdekker and Peter Spreij We consider the intensity-based approach for the modeling of default times of one or more companies. In this approach the default times are defined as the jump times of a Cox process, which is a Poisson process conditional on the realization of its intensity. We assume that the intensity follows the Cox-Ingersoll-Ross model. This model allows one to calculate survival probabilities and prices of defaultable bonds explicitly. In this paper we assume that the Brownian motion, that drives the intensity, is not observed. Using filtering theory for point process observations, we are able to derive dynamics for the intensity and its moment generating function, given the observations of the Cox process. A transformation of the dynamics of the conditional moment generating function allows us to solve the filtering problem, between the jumps of the Cox process, as well as at the jumps. Assuming that the initial distribution of the intensity is of the Gamma type, we obtain an explicit solution to the filtering problem for all t>0. We conclude the paper with the observation that the resulting conditional moment generating function at time t corresponds to a mixture of Gamma distributions. http://arxiv.org/abs/0802.1407 --------------------------------------------------------------- 6680. LAWS OF LARGE NUMBERS FOR EPIDEMIC MODELS WITH COUNTABLY MANY TYPES A.D. Barbour and M.J. Luczak In modelling parasitic diseases, it is natural to distinguish hosts according to the number of parasites that they carry, leading to a countably infinite type space. Proving the analogue of the deterministic equations, used in models with finitely many types as a `law of large numbers' approximation to the underlying stochastic model, has previously either been done case by case, using some special structure, or else not attempted. In this paper, we prove a general theorem of this sort, and complement it with a rate of convergence in the $\ell_1$-norm. http://arxiv.org/abs/0802.1478 --------------------------------------------------------------- 6681. LEARNING NONSINGULAR PHYLOGENIES AND HIDDEN MARKOV MODELS Elchanan Mossel and S\'{e}bastien Roch In this paper we study the problem of learning phylogenies and hidden Markov models. We call a Markov model nonsingular if all transition matrices have determinants bounded away from 0 (and 1). We highlight the role of the nonsingularity condition for the learning problem. Learning hidden Markov models without the nonsingularity condition is at least as hard as learning parity with noise, a well-known learning problem conjectured to be computationally hard. On the other hand, we give a polynomial-time algorithm for learning nonsingular phylogenies and hidden Markov models. http://arxiv.org/abs/cs/0502076 --------------------------------------------------------------- 6682. ON LEARNING THRESHOLDS OF PARITIES AND UNIONS OF RECTANGLES IN RANDOM WALK MODELS S. Roch In a recent breakthrough, [Bshouty et al., 2005] obtained the first passive-learning algorithm for DNFs under the uniform distribution. They showed that DNFs are learnable in the Random Walk and Noise Sensitivity models. We extend their results in several directions. We first show that thresholds of parities, a natural class encompassing DNFs, cannot be learned efficiently in the Noise Sensitivity model using only statistical queries. In contrast, we show that a cyclic version of the Random Walk model allows to learn efficiently polynomially weighted thresholds of parities. We also extend the algorithm of Bshouty et al. to the case of Unions of Rectangles, a natural generalization of DNFs to $\{0,...,b-1\}^n$. http://arxiv.org/abs/cs/0605048 --------------------------------------------------------------- 6683. INCOMPLETE LINEAGE SORTING: CONSISTENT PHYLOGENY ESTIMATION FROM MULTIPLE LOCI Elchanan Mossel and Sebastien Roch We introduce a simple algorithm for reconstructing phylogenies from multiple gene trees in the presence of incomplete lineage sorting, that is, when the topology of the gene trees may differ from that of the species tree. We show that our technique is statistically consistent under standard stochastic assumptions, that is, it returns the correct tree given sufficiently many unlinked loci. We also show that it can tolerate moderate estimation errors. http://arxiv.org/abs/0710.0262 --------------------------------------------------------------- 6684. PHYLOGENIES WITHOUT BRANCH BOUNDS: CONTRACTING THE SHORT, PRUNING THE DEEP Constantinos Daskalakis and Elchanan Mossel and Sebastien Roch We introduce a new phylogenetic reconstruction algorithm which, unlike most previous rigorous inference techniques, does not rely on assumptions regarding the branch lengths or the depth of the tree. The algorithm returns a forest which is guaranteed to contain all edges that are: 1) sufficiently long and 2) sufficiently close to the leaves. How much of the true tree is recovered depends on the sequence length provided. The algorithm is distance- based and runs in polynomial time. http://arxiv.org/abs/0801.4190 --------------------------------------------------------------- 6685. SHRINKAGE EFFECT IN ANCESTRAL MAXIMUM LIKELIHOOD Elchanan Mossel and Sebastien Roch and Mike Steel Ancestral maximum likelihood (AML) is a method that simultaneously reconstructs a phylogenetic tree and ancestral sequences from extant data (sequences at the leaves). The tree and ancestral sequences maximize the probability of observing the given data under a Markov model of sequence evolution, in which branch lengths are also optimized but constrained to take the same value on any edge across all sequence sites. AML differs from the more usual form of maximum likelihood (ML) in phylogenetics because ML averages over all possible ancestral sequences. ML has long been known to be statistically consistent -- that is, it converges on the correct tree with probability approaching 1 as the sequence length grows. However, the statistical consistency of AML has not been formally determined, despite informal remarks in a literature that dates back 20 years. In this short note we prove a general result that implies that AML is statistically inconsistent. In particular we show that AML can `shrink' short edges in a tree, resulting in a tree that has no internal resolution as the sequence length grows. Our results apply to any number of taxa. http://arxiv.org/abs/0802.0914 --------------------------------------------------------------- 6686. PROPERTIES OF THE DENSITY FOR A THREE DIMENSIONAL STOCHASTIC WAVE EQUATION Marta Sanz-Sol\'e We consider a stochastic wave equation in space dimension three driven by a noise white in time and with an absolutely continuous correlation measure given by the product of a smooth function and a Riesz kernel. Let $p_{t,x}(y) $ be the density of the law of the solution $u(t,x)$ of such an equation at points $(t,x)\in]0,T]\times \IR^3$. We prove that the mapping $(t,x)\mapsto p_{t,x}(y)$ owns the same regularity as the sample paths of the process $\{u(t,x), (t,x)\in]0,T]\times \mathbbR^3\}$ established Dalang and Sanz-Sol\'e [Memoirs of the AMS, to appear]. The proof relies on Malliavin calculus and more explicitely, Watanabe's integration by parts formula and estimates derived form it. http://arxiv.org/abs/0802.1607 --------------------------------------------------------------- 6687. ASYMPTOTIC EQUIVALENCE AND CONTIGUITY OF SOME RANDOM GRAPHS Svante Janson We show that asymptotic equivalence, in a strong form, holds between two random graph models with slightly differing edge probabilities under substantially weaker conditions than what might naively be expected. One application is a simple proof of a recent result by van den Esker, van der Hofstad and Hooghiemstra on the equivalence between graph distances for some random graph models. http://arxiv.org/abs/0802.1637 --------------------------------------------------------------- 6688. ASCENDING RUNS IN DEPENDENT UNIFORMLY DISTRIBUTED RANDOM VARIABLES: APPLICATION TO WIRELESS NETWORKS Nathalie Mitton (INRIA Futurs) and Katy Paroux (LM-Besan\c{c}on) and Bruno Sericola (IRISA), S\'ebastien Tixeuil (INRIA Futurs) We analyze in this paper the longest increasing contiguous sequence or maximal ascending run of random variables with common uniform distribution but not independent. Their dependence is characterized by the fact that two successive random variables cannot take the same value. Using a Markov chain approach, we study the distribution of the maximal ascending run and we develop an algorithm to compute it. This problem comes from the analysis of several self-organizing protocols designed for large-scale wireless sensor networks, and we show how our results apply to this domain. http://arxiv.org/abs/0802.1387 --------------------------------------------------------------- 6689. CONVERGENCE OF SOME LEADER ELECTION ALGORITHMS Svante Janson and Christian Lavault and Guy Louchard We start with a set of n players. With some probability P(n,k), we kill n-k players; the other ones stay alive, and we repeat with them. What is the distribution of the number X_n of phases (or rounds) before getting only one player? We present a probabilistic analysis of this algorithm under some conditions on the probability distributions P(n,k), including stochastic monotonicity and the assumption that roughly a fixed proportion alpha of the players survive in each round. We prove a kind of convergence in distribution for X_n-log_a n, where the basis a=1/alpha; as in many other similar problems there are oscillations and no true limit distribution, but suitable subsequences converge, and there is an absolutely continuous random variable Z such that the distribution of X_n can be approximated by Z+log_a n rounded to the nearest larger integer. Applications of the general result include the leader election algorithm where players are eliminated by independent coin tosses and a variation of the leader election algorithm proposed by W.R. Franklin. We study the latter algorithm further, including numerical results. http://arxiv.org/abs/0802.1389 --------------------------------------------------------------- 6690. A TRANSFERENCE METHOD IN QUANTUM PROBABILITY Marius Junge and Javier Parcet Working with a rather general notion of independence, we provide a transference method which allows to compare the p-norm of sums of independent copies with the p-norm of sums of free copies. Our main technique is to construct explicit operator space Lp embeddings preserving independence to reduce the problem to L1, where some recent results by the first-named author can be used. We find applications for noncommutative Khincthine/ Rosenthal type inequalities and for noncommutative Lp embedding theory. http://arxiv.org/abs/0802.1593 --------------------------------------------------------------- 6691. CONDITIONS FOR STABILITY AND INSTABILITY OF RETRIAL QUEUEING SYSTEMS WITH GENERAL RETRIAL TIMES Tewfik Kernane (USTHB) We study the stability of single server retrial queues under general distribution for retrial times and stationary ergodic service times, for three main retrial policies studied in the literature: classical linear, constant and control policies. The approach used is the renovating events approach to obtain sufficient stability conditions by strong coupling convergence of the process modeling the dynamics of the system to a unique stationary ergodic regime. We also obtain instability conditions by convergence in distribution to improper limiting sequences. http://arxiv.org/abs/0802.1812 --------------------------------------------------------------- 6692. MOMENT EXPLOSIONS AND LONG-TERM BEHAVIOR OF AFFINE STOCHASTIC VOLATILITY MODELS Martin Keller-Ressel We consider a class of asset pricing models, where the risk-neutral joint process of log-price and its stochastic variance is an affine process in the sense of Duffie, Filipovic and Schachermayer [2003]. First we obtain conditions for the price process to be conservative and a martingale. Then we present some results on the long-term behavior of the model, including an expression for the invariant distribution of the stochastic variance process. We study moment explosions of the price process, and provide explicit expressions for the time at which a moment of given order becomes infinite. We discuss applications of these results, in particular to the asymptotics of the implied volatility smile, and conclude with some calculations for the Heston model, a model of Bates and the Barndorff-Nielsen-Shephard model. http://arxiv.org/abs/0802.1823 --------------------------------------------------------------- 6693. LEVY-SHEFFER SYSTEMS AND THE LONGSTAFF-SCHWARTZ ALGORITHM FOR AMERICAN OPTION PRICING Stefan Gerhold Glasserman and Yu (Ann. Appl. Probab. 14, 2004, p. 2090) have investigated the mean square error in the Longstaff-Schwartz algorithm for American option pricing, assuming that the underlying process is (geometric) Brownian motion. In this note we provide similar convergence results for the standard Poisson, Gamma, Pascal, and Meixner processes, pointing out the connection of the problem to the L\'evy-Sheffer systems introduced by Schoutens. http://arxiv.org/abs/0802.1831 --------------------------------------------------------------- 6694. ASYMPTOTIC BEHAVIOUR OF RANDOMLY REFLECTING BILLIARDS IN UNBOUNDED TUBULAR DOMAINS Mikhail V. Menshikov and Marina Vachkovskaia and Andrew R. Wade We study stochastic billiards in infinite planar domains with curvilinear boundaries: that is, piecewise deterministic motion with randomness introduced via random reflections at the domain boundary. Physical motivation for the process originates with ideal gas models in the Knudsen regime, with particles reflecting off microscopically rough surfaces. We classify the process into recurrent and transient cases. We also give almost-sure results on the long-term behaviour of the location of the particle, including a super-diffusive rate of escape in the transient case. A key step in obtaining our results is to relate our process to an instance of a one-dimensional stochastic process with asymptotically zero drift, for which we prove some new almost-sure bounds of independent interest. We obtain some of these bounds via an application of general semimartingale criteria, which are again of some independent interest. http://arxiv.org/abs/0802.1865 --------------------------------------------------------------- 6695. MARKOVIAN EMBEDDINGS OF GENERAL RANDOM STRINGS Manuel Lladser Let A be a finite set and X a sequence of A-valued random variables. We do not assume any particular correlation structure between these random variables; in particular, X may be a non-Markovian sequence. An adapted embedding of X is a sequence of the form R(X_1), R(X_1,X_2), R(X_1,X_2,X_3), etc where R is a transformation defined over finite length sequences. In this extended abstract we characterize a wide class of adapted embeddings of X that result in a first-order homogeneous Markov chain. We show that any transformation R has a unique coarsest refinement R' in this class such that R'(X_1), R'(X_1,X_2), R'(X_1,X_2,X_3), etc is Markovian. (By refinement we mean that R'(u)=R'(v) implies R(u)=R(v), and by coarsest refinement we mean that R' is a deterministic function of any other refinement of R in our class of transformations.) We propose a specific embedding that we denote as R^X which is particularly amenable for analyzing the occurrence of patterns described by regular expressions in X. A toy example of a non-Markovian sequence of 0's and 1's is analyzed thoroughly: discrete asymptotic distributions are established for the number of occurrences of a certain regular pattern in X_1,...,X_n, as n tends to infinity, whereas a Gaussian asymptotic distribution is shown to apply for another regular pattern. http://arxiv.org/abs/0802.1896 --------------------------------------------------------------- 6696. ON A THEOREM OF V. BERNIK IN THE METRICAL THEORY OF DIOPHANTINE APPROXIMATION Victor Beresnevich This paper goes back to a famous problem of Mahler in metrical Diophantine approximation. The problem has been settled by Sprindzuk and subsequently improved by Alan Baker and Vasili Bernik. In particular, Bernik's result establishes a convergence Khintchine type theorem for Diophantine approximation by polynomials, that is it allows arbitrary monotonic error of approximation. In the present paper the monotonicity assumption is completely removed. http://arxiv.org/abs/0802.1910 --------------------------------------------------------------- 6697. DIFFEOMORPHISMS OF THE CIRCLE AND BROWNIAN MOTIONS ON AN INFINITE-DIMENSIONAL SYMPLECTIC GROUP Maria Gordina and Mang Wu An embedding of the group $\Diff(S^{1})$ of orientation preserving diffeomorphims of the unit circle $S^1$ into an infinite-dimensional symplectic group, $\Sp(\infty)$, is studied. The authors prove that this embedding is not surjective. A Brownian motion is constructed on $\Sp(\infty)$. This study is motivated by recent work of H. Airault, S. Fang and P. Malliavin. http://arxiv.org/abs/0802.1955 --------------------------------------------------------------- 6698. ON THE KERT\'ESZ LINE: SOME RIGOROUS BOUNDS Jean Ruiz (CPT) and Marc Wouts (MODAL'x) We study the Kert\'esz line of the $q$--state Potts model at (inverse) temperature $\beta$, in presence of an external magnetic field $h$. This line separates two regions of the phase diagram according to the existence or not of an infinite cluster in the Fortuin-Kasteleyn representation of the model. It is known that the Kert\'esz line $h_K (\beta)$ coincides with the line of first order phase transition for small fields when $q$ is large enough. Here we prove that the first order phase transition implies a jump in the density of the infinite cluster, hence the Kert\'esz line remains below the line of first order phase transition. We also analyze the region of large fields and prove, using techniques of stochastic comparisons, that $h_K (\beta)$ equals $ \log (q - 1) - \log (\beta - \beta_p)$ to the leading order, as $\beta$ goes to $\beta_p = - \log (1 - p_c)$ where $p_c$ is the threshold for bond percolation. http://arxiv.org/abs/0802.1826 --------------------------------------------------------------- 6699. NON-HOMOGENEOUS POLYGONAL MARKOV FIELDS IN THE PLANE: GRAPHICAL REPRESENTATIONS AND GEOMETRY OF HIGHER ORDER CORRELATIONS Tomasz Schreiber We consider polygonal Markov fields originally introduced by Arak and Surgailis (1989). Our attention is focused on fields with nodes of order two, which can be regarded as continuum ensembles of non-intersecting contours in the plane, sharing a number of features with the two-dimensional Ising model. We introduce non-homogeneous version of polygonal fields in anisotropic enviroment. For these fields we provide a class of new graphical constructions and random dynamics. These include a generalised dynamic representation, generalised and defective disagreement loop dynamics as well as a generalised contour birth and death dynamics. Next, we use these constructions as tools to obtain new exact results on the geometry of higher order correlations of polygonal Markov fields in their consistent regime. http://arxiv.org/abs/0802.2115 --------------------------------------------------------------- 6700. META-STABILITY AND CONDENSED ZERO-RANGE PROCESSES ON FINITE SETS J. Beltran and C. Landim We propose a definition o meta-stability and obtain sufficient conditions for a sequence of Markov processes on finite state spaces to be meta- stable. In the reversible case, these conditions reduce to estimates of the capacity and the measure of certain meta-stable sets. We prove that a class of condensed zero-range processes with asymptotically decreasing jump rates is meta- stable. http://arxiv.org/abs/0802.2171 --------------------------------------------------------------- 6701. EFFICIENT HEDGING AND RISK MINIMIZATION Marie-Amelie Morlais In that paper, we solve dynamically a partial hedging problem for an American contingent claim: assuming superhedging is not feasible, we explain in this context the notion of efficient hedging by introducing a risk minimization criterion: we consider here the problem of minimizing the conditional expected loss for a given convex and non decreasing loss function. To solve this problem, we provide a connection between the dynamic convex risk functional introduced and the solution of a quadratic RBSDE (Reflected Backward Stochastic Differential Equations): this is achieved by studying the properties of specific non linear expectations. http://arxiv.org/abs/0802.2172 --------------------------------------------------------------- 6702. EXPLICIT PARAMETRIX AND LOCAL LIMIT THEOREMS FOR SOME DEGENERATE DIFFUSION PROCESSES Valentin Konakov (CMI RAS) and Stephane Menozzi (PMA) and Stanislav Molchanov For a class of degenerate diffusion processes of rank 2, i.e. when only Poisson brackets of order one are needed to span the whole space, we obtain a parametrix representation of the density from which we derive some explicit Gaussian controls that characterize the additional singularity induced by the degeneracy. We then give a local limit theorem with the usual convergence rate for an associated Markov chain approximation. The key point is that the "weak" degeneracy allows to exploit the techniques first introduced by Konakov and Molchanov and then developed by Konakov and Mammen that rely on Gaussian approximations. http://arxiv.org/abs/0802.2229 --------------------------------------------------------------- 6703. CYLINDRICAL WIENER PROCESSES Markus Riedle In this work cylindrical Wiener processes on Banach spaces are defined by means of cylindrical stochastic processes, which are a well considered mathematical object. This approach allows a definition which is a simple straightforward extension of the real-valued situation. We apply this definition to introduce a stochastic integral with respect to cylindrical Wiener processes. Again, this definition is a straightforward extension of the real-valued situation which results now in simple conditions on the integrand. In particular, we do not have to put any geometric constraints on the Banach space under consideration. Finally, we relate this integral to well- known stochastic integrals in literature. http://arxiv.org/abs/0802.2261 --------------------------------------------------------------- 6704. RATIONAL FUNCTIONS ASSOCIATED WITH THE WHITE NOISE SPACE AND RELATED TOPICS Daniel Alpay and David Levanony Motivated by the hyper-holomorphic case we introduce and study rational functions in the setting of Hida's white noise space. The Fueter polynomials are replaced by a basis computed in terms of the Hermite functions, and the Cauchy-Kovalevskaya product is replaced by the Wick product. http://arxiv.org/abs/0802.2373 --------------------------------------------------------------- 6705. MARKOV LOOPS AND RENORMALIZATION Yves Le Jan (LM-Orsay) We study Poissonnian ensembles of Markov loops and the associated renormalized self-intersection local times. http://arxiv.org/abs/0802.2478 --------------------------------------------------------------- 6706. A GENERAL BALLOT THEOREM L. Addario-Berry and B.A. Reed We prove an analogue of the classical ballot theorem that holds for any random walk in the range of attraction of the normal distribution. Our result is best possible: we exhibit examples demonstrating that if any of our hypotheses are removed, our conclusions may no longer hold. http://arxiv.org/abs/0802.2491 --------------------------------------------------------------- 6707. CONSTRUCTION OF A STATIONARY FIFO QUEUE WITH IMPATIENT CUSTOMERS Pascal Moyal In this paper, we study the stability of queues with impatient customers. Under general stationary ergodic assumptions, we first provide some conditions for such a queue to be regenerative (i.e. to empty a.s. an infinite number of times). In the particular case of a single server operating in First in, First out, we prove the existence (in some cases, on an enlarged probability space) of a stationary workload. This is done by studying stochastic recursions under the Palm settings, and by stochastic comparison of stochastic recursions. http://arxiv.org/abs/0802.2495 --------------------------------------------------------------- 6708. NON COMMUTATIVE CONDITIONAL EXPECTATIONS, PREDICTION AND A NEW LOOK AT SOME QUANTUM PARADOXES Henryk Gzyl When the result of an observation is taken into account by means of a non-commutative conditional expectation, exactly as in classical prediction theory, some of the usual paradoxes cease to be so. The moral of this note is that the mystery of the probabilistic interpretation of quantum mechanics lies in the superposition principle http://arxiv.org/abs/0802.2297 --------------------------------------------------------------- 6709. ON THE SHUFFLING ALGORITHM FOR DOMINO TILINGS Eric Nordenstam We study the dynamics of a certain discrete model of interacting particles that comes from the so called shuffling algorithm for sampling a random tiling of an Aztec diamond. It turns out that the transition probabilities have a particularly convenient determinantal form. An analogous formula in a continuous setting has recently been obtained by Jon Warren studying certain model of interlacing Brownian motions which can be used to construct Dyson's non-intersecting Brownian motion. We conjecture that Warren's model can be recovered as a scaling limit of our discrete model and prove some partial results in this direction. As an application to one of these results we use it to rederive the known result that random tilings of an Aztec diamond, suitably rescaled near a turning point, converge to the GUE minor process. http://arxiv.org/abs/0802.2592 --------------------------------------------------------------- 6710. COUPLING-CUTOFFS FOR RANDOM WALKS ON THE HYPERCUBE Stephen Connor We consider a simple independence coupling for two continuous-time random walks on the hypercube, and investigate when the tail probability of the coupling time exhibits `cutoff behaviour'. We not only provide a necessary and sufficient condition for this so-called `coupling-cutoff' to occur, but also prove a general bound on the window size of the cutoff, making use of the Lambert W-function. The results may be generalised to n-tuples of independent Markov processes for which each component may be coupled at an exponential rate. http://arxiv.org/abs/0802.2641 --------------------------------------------------------------- 6711. COMPLETE MOMENT AND INTEGRAL CONVERGENCE FOR SUMS OF NEGATIVELY ASSOCIATED RANDOM VARIABLES Han-Ying Liang and Deli Li and Andrew Rosalsky For a sequence of identically distributed negatively associated random variables $\{X_n; n\geq 1\}$ with partial sums $S_n=\sum_{i=1}^nX_i, n \geq 1$, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form $$ \sum_{n \ge n_0} n^{r -2 -\frac{1}{pq}} a_n E(\max_{1 \le k \le n}|S_k| ^{\frac{1}{q}} - \epsilon b_n^{\frac{1}{pq}})^+ < \infty $$ to hold where $r>1, q>0$ and either $n_0=1, 0 R with Laplacian interaction of the form \sum_i V(\Delta \phi_i), where \Delta is the discrete Laplacian and the potential V(.) is symmetric and uniformly strictly convex. The pinning model is defined by giving the field a reward \epsilon \ge 0 each time it touches the x-axis, that plays the role of a defect line. It is known that this model exhibits a phase transition between a delocalized regime (\epsilon < \epsilon_c) and a localized one (\epsilon > \epsilon_c), where 0 < \epsilon_c < \infty. In this paper we give a precise pathwise description of the transition, extracting the full scaling limits of the model. We show in particular that in the delocalized regime the field wanders away from the defect line at a typical distance N^{3/2}, while in the localized regime the distance is just O((log N)^2). A subtle scenario shows up in the critical regime (\epsilon = \epsilon_c), where the field, suitably rescaled, converges in distribution toward the derivative of a symmetric stable Levy process of index 2/5. Our approach is based on Markov renewal theory. http://arxiv.org/abs/0802.3154 --------------------------------------------------------------- 6721. HARMONIC MEASURE AND WINDING OF RANDOM CONFORMAL PATHS: A COULOMB GAS PERSPECTIVE Bertrand Duplantier and Ilia Binder We consider random conformally invariant paths in the complex plane (SLEs). Using the Coulomb gas method in conformal field theory, we rederive the mixed multifractal exponents associated with both the harmonic measure and winding (rotation or monodromy) near such critical curves, previously obtained by quantum gravity methods. The results also extend to the general cases of harmonic measure moments and winding of multiple paths in a star configuration. http://arxiv.org/abs/0802.2280 --------------------------------------------------------------- 6722. RELATIONSHIP BETWEEN STOCHASTIC FLOWS AND CONNECTION FORMS M. Neklyudov In this article I will prove new representation for the Levi-Civita connection in terms of the stochastic flow corresponding to Brownian motion on manifold. http://arxiv.org/abs/0802.3255 --------------------------------------------------------------- 6723. A MODEL OF CONTINUOUS TIME POLYMER ON THE LATTICE David Marquez-Carreras and Carles Rovira and Samy Tindel In this article, we try to give a rather complete picture of the behavior of the free energy for a model of directed polymer in a random environment, in which the polymer is a simple symmetric random walk on the lattice $ \Z^d$, and the environment is a collection $\{W(t,x);t\ge 0, x\in \Z^d\}$ of i.i.d. Brownian motions. http://arxiv.org/abs/0802.3296 --------------------------------------------------------------- 6724. ASYMPTOTIC BEHAVIOR OF WEIGHTED QUADRATIC VARIATIONS OF FRACTIONAL BROWNIAN MOTION: THE CRITICAL CASE H=1/4 Ivan Nourdin (PMA) and Anthony R\'eveillac (LMA) We derive the asymptotic behavior of weighted quadratic variations of fractional Brownian motion $B$ with Hurst index H=1/4. This completes the only missing case in a very recent work by I. Nourdin, D. Nualart and C.A. Tudor. Moreover, as an application, we solve a recent conjecture of K. Burdzy and J. Swanson on the asymptotic behavior of the Riemann sums with alternating signs associated to B. http://arxiv.org/abs/0802.3307 --------------------------------------------------------------- 6725. VARIATIONS OF THE SOLUTION TO A STOCHASTIC HEAT EQUATION II Krzysztof Burdzy and Jason Swanson We consider the solution u(x,t) to a stochastic heat equation. For fixed x, the process F(t) = u(x,t) has a nontrivial quartic variation. It follows that F is not a semimartingale, so a stochastic integral with respect to F cannot be defined in the classical Ito sense. We show that for sufficiently differentiable functions g, a stochastic integral \int g(F) dF exists as a limit of discrete, midpoint style Riemann sums, where the limit is taken in distribution in the Skorohod space of cadlag functions. Moreover, we show that this integral satisfies a change of variables formulas with a correction term that is an ordinary Ito integral with respect to a Brownian motion that is independent of F. http://arxiv.org/abs/0802.3356 --------------------------------------------------------------- 6726. THE SUBELLIPTIC HEAT KERNEL ON SU(2): REPRESENTATIONS, ASYMPTOTICS AND GRADIENT BOUNDS Fabrice Baudoin and Michel Bonnefont The Lie group SU(2) endowed with its canonical subriemannian structure appears as a three-dimensional model of a positively curved} subelliptic space. The goal of this work is to study the subelliptic heat kernel on it and some related functional inequalities. http://arxiv.org/abs/0802.3320 --------------------------------------------------------------- 6727. MOMENT PROBLEMS AND BOUNDARIES OF NUMBER TRIANGLES Alexander Gnedin and Jim Pitman The boundary problem for graphs like Pascal's but with general multiplicities of edges is related to a `backward' problem of moments of the Hausdorff type. http://arxiv.org/abs/0802.3410 --------------------------------------------------------------- 6728. ON A NONHIERARCHICAL VERSION OF THE GENERALIZED RANDOM ENERGY MODEL. II. ULTRAMETRICITY Erwin Bolthausen and Nicola Kistler We study the Gibbs measure of the nonhierarchical versions of the Generalized Random Energy Models introduced in previous work, [2]. We prove that the ultrametricity holds only provided some nondegeneracy conditions on the hamiltonian are met. http://arxiv.org/abs/0802.3436 --------------------------------------------------------------- 6729. A TRUNCATION APPROACH FOR FAST COMPUTATION OF DISTRIBUTION FUNCTIONS Xinjia Chen In this paper, we propose a general approach for improving the efficiency of computing distribution functions. The idea is to truncate the domain of summation or integration. http://arxiv.org/abs/0802.3455 --------------------------------------------------------------- 6730. THE AIZENMAN-SIMS-STARR AND GUERRA'S SCHEMES FOR THE SK MODEL WITH MULTIDIMENSIONAL SPINS Anton Bovier and Anton Klimovsky We prove upper and lower bounds on the free energy in the Sherrington-Kirkpatrick model with multidimensional (e.g., Heisenberg) spins in terms of the variational inequalities based on the corresponding Parisi functional. We employ the comparison scheme of Aizenman, Sims and Starr and the one of Guerra involving the generalised random energy model-inspired processes and Ruelle's probability cascades. For this purpose an abstract quenched large deviations principle of the Gaertner-Ellis type is obtained. Using the properties of Ruelle's probability cascades and the Bolthausen-Sznitman coalescent, we derive Talagrand's representation of the Guerra remainder term for our model. We study the properties of the multidimensional Parisi functional by establishing a link with a certain class of the non-linear partial differential equations. Solving a problem posed by Talagrand, we show the strict convexity of the local Parisi functional. We prove the Parisi formula for the local free energy in the case of the multidimensional Gaussian a priori distribution of spins using Talagrand's methodology of the a priori estimates. http://arxiv.org/abs/0802.3467 --------------------------------------------------------------- 6731. THE WAITING TIME FOR M MUTATIONS Jason Schweinsberg We consider a model of a population of fixed size N in which each individual gets replaced at rate one and each individual experiences a mutation at rate \mu. We calculate the asymptotic distribution of the time that it takes before there is an individual in the population with m mutations. Several different behaviors are possible, depending on how \mu changes with N. These results have applications to the problem of determining the waiting time for regulatory sequences to appear and to models of cancer development. http://arxiv.org/abs/0802.3485 --------------------------------------------------------------- 6732. RECONSTRUCTION OF RANDOM COLOURINGS Allan Sly Reconstruction problems have been studied in a number of contexts including biology, information theory and and statistical physics. We consider the reconstruction problem for random $k$-colourings on the $\Delta$-ary tree for large $k$. Bhatnagar et. al. showed non-reconstruction when $\Delta \leq \frac12 k\log k - o(k\log k)$ and reconstruction when $\Delta \geq k \log k + o(k\log k)$. We tighten this result and show non-reconstruction when $ \Delta \leq k[\log k + \log \log k + 1 - \ln 2 -o(1)]$ and reconstruction when $\Delta \geq k[\log k + \log \log k + 1+o(1)]$. http://arxiv.org/abs/0802.3487 --------------------------------------------------------------- 6733. THE TIME CONSTANT VANISHES ONLY ON THE PERCOLATION CONE IN DIRECTED FIRST PASSAGE PERCOLATION Yu Zhang We consider the directed first passage percolation model on ${\bf Z}^2$. In this model, we assign independently to each edge $e$ a passage time $t(e)$ with a common distribution $F$. We denote by $\vec{T}({\bf 0}, (r,\theta))$ the passage time from the origin to $(r, \theta)$ by a northeast path for $ (r, \theta)\in {\bf R}^+\times [0,\pi/2]$. It is known that $\vec{T}({\bf 0}, (r, \theta))/r$ converges to a time constant $\vec{\mu}_F (\theta)$. Let $\vec{p}_c$ denote the critical probability for oriented percolation. In this paper, we show that the time constant has a phase transition divided by $\vec{p}_c$, as follows: (1) If $F(0) < \vec{p}_c$, then $\vec{\mu}_F(\theta) >0$ for all $0\leq \theta\leq \pi/2$. (2) If $F(0) = \vec{p}_c$, then $\vec{\mu}_F(\theta) >0$ if and only if $\theta\neq \pi/4$. (3) If $F(0)=p > \vec{p}_c$, then there exists a percolation cone between $\theta_p^-$ and $\theta_p^+$ for $0\leq \theta^-_p< \theta^+_p \leq \pi/2$ such that $\vec{\mu} (\theta) >0$ if and only if $\theta\not\in [\theta_p^-, \theta^+_p]$. Furthermore, all the moments of $\vec{T}({\bf 0}, (r, \theta))$ converge whenever $\theta\in [\theta_p^-, \theta^+_p]$. As applications, we describe the shape of the directed growth model on the distribution of $F$. We give a phase transition for the shape divided by $\vec{p}_c$. http://arxiv.org/abs/0802.3519 --------------------------------------------------------------- 6734. ON EQUILIBRIUM PRICES IN CONTINUOUS TIME V. Filipe Martins-da-Rocha and Frank Riedel We combine general equilibrium theory and theorie generale of stochastic processes to derive structural results about equilibrium state prices. http://arxiv.org/abs/0802.3585 --------------------------------------------------------------- 6735. VERAVERBEKE'S THEOREM AT LARGE - ON THE MAXIMUM OF SOME PROCESSES WITH NEGATIVE DRIFT AND HEAVY TAIL INNOVATIONS Philippe Barbe (CNRS) and Bill McCormick (UGA) Veraverbeke's (1977) theorem relates the tail of the distribution of the supremum of a random walk with negative drift to the tail of the distribution of its increments, or equivalently, the probability that a centered random walk with heavy-tail increments hits a moving linear boundary. We study similar problems for more general processes. In particular, we derive an analogue of Veraverbeke's theorem for fractional integrated ARMA models without prehistoric influence, when the innovations have regularly varying tails. Furthermore, we prove some limit theorems for the trajectory of the process, conditionally on a large maximum. Those results are obtained by using a general scheme of proof which we present in some detail and should be of value in other related problems. http://arxiv.org/abs/0802.3638 --------------------------------------------------------------- 6736. RANDOM WALK ON A DISCRETE TORUS AND RANDOM INTERLACEMENTS David Windisch We investigate the relation between the local picture left by the trajectory of a simple random walk on the torus (Z/NZ)^d, d >= 3, until u N^d time steps, u > 0, and the model of random interlacements recently introduced by Sznitman. In particular, we show that for large N, the joint distribution of the local pictures in the neighborhoods of finitely many distant points left by the walk up to time u N^d converges to independent copies of the random interlacement at level u. http://arxiv.org/abs/0802.3654 --------------------------------------------------------------- 6737. A SIMPLE FORMULA FOR CONSTRUCTING CONFIDENCE INTERVAL FOR THE MEAN OF BOUNDED RANDOM VARIABLES Xinjia Chen In this article, we derive an explicit formula for computing confidence interval for the mean of bounded random variables. In additional to its simplicity, the formula is very tight in comparison with existing results in literature. http://arxiv.org/abs/0802.3458 --------------------------------------------------------------- 6738. INTERVAL ESTIMATION OF BOUNDED VARIABLE MEANS VIA INVERSE SAMPLING Xinjia Chen In this paper, we develop interval estimation methods for means of bounded random variables based on a sequential procedure such that the sampling is continued until the sample sum is no less than a prescribed threshold. http://arxiv.org/abs/0802.3539 --------------------------------------------------------------- 6739. ON THE LOCALITY OF THE PR\"UFER CODE Craig Lennon The Pr\"ufer code is a bijection between trees on the vertex set $[n]$ and strings on the set $[n]$ of length $n-2$ (Pr\"ufer strings of order $n $). In this paper we examine the `locality' properties of the Pr\"ufer code, i.e. the effect of changing an element of the Pr\"ufer string on the structure of the corresponding tree. Our measure for the distance between two trees $T,T^*$ is $\Delta(T,T^*)=n-1-| E(T)\cap E(T^*)|$. We randomly mutate the $\mu$th element of the Pr\"ufer string of the tree $T$, changing it to the tree $T^*$, and we asymptotically estimate the probability that this results in a change of $\ell$ edges, i.e. $P(\Delta=\ell | \mu).$ We find that P(\Delta=\ell | \mu)$ is on the order of $ n^{-1/3+o(1)}$ for any integer $\ell>1,$ and that $P(\Delta=1 | \mu)=(1-\mu/n)^2+o(1).$ This result implies that the probability of a `perfect' mutation in the Pr\"ufer code (one for which $\Delta(T,T^*)=1$) is $1/3.$ http://arxiv.org/abs/0802.3514 --------------------------------------------------------------- 6740. STRONG SOLUTIONS FOR STOCHASTIC POROUS MEDIA EQUATIONS WITH JUMPS Viorel Barbu and Carlo Marinelli We prove global well-posedness in the strong sense for stochastic generalized porous media equations driven by locally square integrable martingales with stationary independent increments. http://arxiv.org/abs/0802.3594 --------------------------------------------------------------- 6741. ASYMPTOTICALLY OPTIMAL QUANTIZATION SCHEMES FOR GAUSSIAN PROCESSES Harald Luschgy and Gilles Pag\`es (PMA) and Benedikt Wilbertz We describe quantization designs which lead to asymptotically and order optimal functional quantizers. Regular variation of the eigenvalues of the covariance operator plays a crucial role to achieve these rates. For the development of a constructive quantization scheme we rely on the knowledge of the eigenvectors of the covariance operator in order to transform the problem into a finite dimensional quantization problem of normal distributions. Furthermore we derive a high-resolution formula for the $L^2$- quantization errors of Riemann-Liouville processes. http://arxiv.org/abs/0802.3761 --------------------------------------------------------------- 6742. STOCHASTIC TAMED 3D NAVIER-STOKES EQUATIONS: EXISTENCE, UNIQUENESS AND ERGODICITY Michael R\"ockner and Xicheng Zhang In this paper, we prove the existence of a unique strong solution to a stochastic tamed 3D Navier-Stokes equation in the whole space as well as in the periodic boundary case. Then, we also study the Feller property of solutions, and prove the existence of invariant measures for the corresponding Feller semigroup in the case of periodic conditions. Moreover, in the case of periodic boundary and degenerated additive noise, using the notion of asymptotic strong Feller property proposed by Hairer and Mattingly \cite{Ha-Ma}, we prove the uniqueness of invariant measures for the corresponding transition semigroup. http://arxiv.org/abs/0802.3934 --------------------------------------------------------------- 6743. A LINK BETWEEN BINOMIAL PARAMETERS AND MEANS OF BOUNDED RANDOM VARIABLES Xinjia Chen In this paper, we establish a fundamental connection between binomial parameters and means of bounded random variables. Such connection finds applications in statistical inference of means of bounded variables. http://arxiv.org/abs/0802.3946 --------------------------------------------------------------- 6744. THE SMALLEST SINGULAR VALUE OF A RANDOM RECTANGULAR MATRIX Mark Rudelson and Roman Vershynin We prove an optimal estimate on the smallest singular value of a random subgaussian matrix, valid for all fixed dimensions. For an N by n matrix A with independent and identically distributed subgaussian entries, the smallest singular value of A is at least of the order \sqrt{N} - \sqrt{n-1} with high probability. A sharp estimate on the probability is also obtained. http://arxiv.org/abs/0802.3956 --------------------------------------------------------------- 6745. EXIT PROBLEM OF A TWO-DIMENSIONAL RISK PROCESS FROM THE QUADRANT: EXACT AND ASYMPTOTIC RESULTS Florin Avram and Zbigniew Palmowski and Martijn Pistorius Consider two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions. We model the occurrence of claims according to a renewal process. One ruin problem considered is that of the corresponding two-dimensional risk process first leaving the positive quadrant; another is that of entering the negative quadrant. When the claims arrive according to a Poisson process we obtain a closed form expression for the ultimate ruin probability. In the general case we analyze the asymptotics of the ruin probability when the initial reserves of both companies tend to infinity under a Cram\'er light-tail assumption on the claim size distribution. http://arxiv.org/abs/0802.4060 --------------------------------------------------------------- 6746. GOOD DEAL BOUNDS INDUCED BY SHORTFALL RISK Takuji Arai We shall provide in this paper good deal pricing bounds for contingent claims induced by the shortfall risk with some loss function. Assumptions we impose on loss functions and contingent claims are very mild. We prove that the upper and lower bounds of good deal pricing bounds are expressed by convex risk measures on Orlicz hearts. In addition, we obtain its representation with the minimal penalty function. Moreover, we give a representation, for two simple cases, of good deal bounds and calculate the optimal strategies when a claim is traded at the upper or lower bounds of its good deal pricing bound. http://arxiv.org/abs/0802.4141 --------------------------------------------------------------- 6747. ORBIT MEASURES AND INTERLACED DETERMINANTAL POINT PROCESSES Manon Defosseux (PMA) We study some random interlaced configurations considering the eigenvalues of the main minors of Hermitian random matrices of the classical complex Lie algebras. We claim that these random configurations are determinantal and give their correlation kernels. http://arxiv.org/abs/0802.4183 --------------------------------------------------------------- 6748. THE HEIGHT OF WATERMELONS WITH WALL Thomas Feierl We derive asymptotics for the moments as well as the weak limit of the height distribution of watermelons with p branches with wall. This generalises a famous result of de Bruijn, Knuth and Rice on the average height of planted plane trees, and results by Fulmek and Katori et al. on the expected value, respectively the higher moments, of the height distribution of watermelons with two branches. The asymptotics for the moments depend on the analytic behaviour of certain multidimensional Dirichlet series. In order to obtain this information we prove a reciprocity relation satisfied by the derivatives of one of Jacobi's theta functions, which generalises the well known reciprocity law for Jacobi's theta functions. http://arxiv.org/abs/0802.2691 --------------------------------------------------------------- 6749. LIMITS LAWS FOR GEOMETRIC MEANS OF FREE POSITIVE RANDOM VARIABLES Gabriel H. Tucci Let $\{a_{k}\}_{k=1}^{\infty}$ be free identically distributed positive non--commuting random variables with probability measure distribution $ \mu$. In this paper we proved a multiplicative version of the Free Central Limit Theorem. More precisely, let $b_{n}=a_{1}^{1/2}a_{2}^{1/2}... a_{n}... a_{2}^{1/2}a_{1}^{1/2}$ then $b_{n}$ is a positive operator with the same moments as $x_{n}=a_{1}a_{2}... a_{n}$ and $b_{n}^{1/2n}$ converges in distribution to positive operator $\Lambda$. We completely determined the probability measure distribution $\nu$ of $\Lambda$ from the distribution $\mu$. This gives us a natural map $\mathcal{G}:\mathcal{M_{+}}\to \mathcal{M_{+}}$ with $\mu\mapsto \mathcal{G}(\mu)=\nu.$ We study how this map behaves with respect to additive and multiplicative free convolution. As an interesting consequence of our results, we illustrate the relation between the probability distribution $\nu$ and the distribution of the Lyapunov exponents for the sequence $\{a_{k}\}_{k=1}^{\infty}$ introduced in \cite{LyaV}. http://arxiv.org/abs/0802.4226 --------------------------------------------------------------- 6750. MULTISTEP BAYESIAN STRATEGY IN COIN-TOSSING GAMES AND ITS APPLICATION TO ASSET TRADING GAMES IN CONTINUOUS TIME Kei Takeuchi and Masayuki Kumon and Akimichi Takemura We study multistep Bayesian betting strategies in coin-tossing games in the framework of game-theoretic probability of Shafer and Vovk (2001). We show that by a countable mixture of these strategies, a gambler or an investor can exploit arbitrary patterns of deviations of nature's moves from independent Bernoulli trials. We then apply our scheme to asset trading games in continuous time and derive the exponential growth rate of the investor's capital when the variation exponent of the asset price path deviates from two. http://arxiv.org/abs/0802.4311 --------------------------------------------------------------- 6751. POSITIVE STOCHASTIC VOLATILITY SIMULATION William Halley and Simon J.A. Malham and Anke Wiese We present a positivity preserving numerical scheme for the pathwise solution of nonlinear stochastic differential equations driven by a multi- dimensional Wiener process and governed by non-commutative linear and non- Lipschitz vector fields. This strong order one scheme uses: (i) Strang exponential splitting, an approximation that decomposes the stochastic flow separately into the drift flow, and the pure diffusion flow governed by the diffusion vector fields; (ii) an implicit Euler method to approximate the drift flow; and (iii) an implicit Milstein method to approximate the pure diffusion flow. The separate approximations for the drift and pure diffusion flows preserve positivity. Therefore the Strang exponential splitting approximation does also. We demonstrate the efficacy of our method by applying it to the Heston model and a variance curve model, and compare it against well-established positivity preserving schemes. http://arxiv.org/abs/0802.4411 --------------------------------------------------------------- 6752. ASYMPTOTIC ANALYSIS OF A FLUID MODEL MODULATED BY AN $M/M/1$ QUEUE Charles Knessl and Diego Dominici We analyze asymptotically a differential-difference equation, that arises in a Markov-modulated fluid model. We use singular perturbation methods to analyze the problem with appropriate scalings of the two state variables. In particular, the ray method and asymptotic matching are used. http://arxiv.org/abs/0802.4434 From pas at lists.imstat.org Sun May 4 16:03:13 2008 From: pas at lists.imstat.org (Probability Abstract Service) Date: Sun, 4 May 2008 23:03:13 +0200 Subject: [PAS] Probability Abstracts 103 Message-ID: <393646FE-CDF2-4468-9F34-0599692E916F@unimi.it> Probability Abstracts 103 This document contains abstracts 6753-6993 from March-1-2008 to April-30-2008. They have been mailed on May 4th, 2008. This letter can be also found on line at http://pas.imstat.org/Letters/letter_103.shtml --------------------------------------------------------------- 6753. THE CONDITIONED RECONSTRUCTED PROCESS Tanja Gernhard We investigate a neutral model for speciation and extinction, the constant rate birth-death process. The process is conditioned to have $n$ extant species today, we look at the tree distribution of the reconstructed trees-- i.e. the trees without the extinct species. Whereas the tree shape distribution is well-known and actually the same as under the pure birth process, no analytic results for the speciation times were known. We provide the distribution for the speciation times and calculate the expectations analytically. This characterizes the reconstructed trees completely. We will show how the results can be used to date phylogenies. http://arxiv.org/abs/0803.0153 --------------------------------------------------------------- 6754. THE SQUARE NEGATIVE CORRELATION PROPERTY FOR GENERALIZED ORLICZ BALLS Jakub Onufry Wojtaszczyk Antilla, Ball and Perissinaki proved that the squares of coordinate functions in $\ell_p^n$ are negatively correlated. This paper extends their results to balls in generalized Orlicz norms on R^n. From this, the concentration of the Euclidean norm and a form of the Central Limit Theorem for the generalized Orlicz balls is deduced. Also, a counterexample for the square negative correlation hypothesis for 1-symmetric bodies is given. Currently the CLT is known in full generality for convex bodies (see the paper "Power-law estimates for the central limit theorem for convex sets" by B. Klartag), while for generalized Orlicz balls a much more general result is true (see "The negative association property for the absolute values of random variables equidistributed on a generalized Orlicz ball" by M. Pilipczuk and J. O. Wojtaszczyk). While, however, both aforementioned papers are rather long, complicated and technical, this paper gives a simple and elementary proof of, eg., the Euclidean concentration for generalized Orlicz balls. http://arxiv.org/abs/0803.0433 --------------------------------------------------------------- 6755. THE NEGATIVE ASSOCIATION PROPERTY FOR THE ABSOLUTE VALUES OF RANDOM VARIABLES EQUIDISTRIBUTED ON A GENERALIZED ORLICZ BALL Marcin Pilipczuk and Jakub Onufry Wojtaszczyk Random variables equidistributed on convex bodies have received quite a lot of attention in the last few years. In this paper we prove the negative association property (which generalizes the subindependence of coordinate slabs) for generalized Orlicz balls. This allows us to give a strong concentration property, along with a few moment comparison inequalities. Also, the theory of negatively associated variables is being developed in its own right, which allows us to hope more results will be available. Moreover, a simpler proof of a more general result for $\ell_p^n$ balls is given. http://arxiv.org/abs/0803.0434 --------------------------------------------------------------- 6756. INDIVIDUAL RISK AND LEBESGUE EXTENSION WITHOUT AGGREGATE UNCERTAINTY Yeneng Sun and Yongchao Zhang Many economic models include random shocks imposed on a large number (continuum) of economic agents with individual risk. In this context, an exact law of large numbers and its converse is presented in Sun [Journal of Economic Theory 126(2006), 31-69] to characterize the cancelation of individual risk via aggregation. However, it is well known that the Lebesgue unit interval is not suitable for modeling a continuum of agents in the particular setting. The purpose of this paper is to show that an extension of the Lebesgue unit interval does work well as an agent space with various desirable properties associated with individual risk. http://arxiv.org/abs/0803.0442 --------------------------------------------------------------- 6757. STEIN'S METHOD AND EXACT BERRY-ESS\'EEN ASYMPTOTICS FOR FUNCTIONALS OF GAUSSIAN FIELDS Ivan Nourdin (PMA) and Giovanni Peccati (LSTA) We show how to detect optimal Berry-Ess\'een bounds in the normal approximation of functionals of Gaussian fields. Our techniques are based on a combination of Malliavin calculus, Stein's method and the method of moments and cumulants, and provide de facto local (one term) Edgeworth expansions. The findings of the present paper represent a further refinement of the main results proved in Nourdin and Peccati (2007b). Among several examples, we discuss three crucial applications: (i) to Toeplitz quadratic functionals of continuous-time stationary processes (extending results by Ginovyan (1994) and Ginovyan and Sahakyan (2007)), (ii) to ``exploding'' quadratic functionals of a Brownian sheet, and (iii) to a continuous-time version of the Breuer- Major CLT for functionals of a fractional Brownian motion. http://arxiv.org/abs/0803.0458 --------------------------------------------------------------- 6758. ROUGH EVOLUTION EQUATIONS Massimiliano Gubinelli and Samy Tindel We show how to generalize Lyons' rough paths theory in order to give a pathwise meaning to some nonlinear infinite-dimensional evolution equations associated to an analytic semigroup and driven by an irregular noise. As an illustration, we apply the theory to a class of 1d SPDEs driven by a space-time fractional Brownian motion. http://arxiv.org/abs/0803.0552 --------------------------------------------------------------- 6759. SUPERPOSITION RULES AND STOCHASTIC LIE-SCHEFFERS SYSTEMS Joan-Andreu L\'azaro-Cam\'i and Juan-Pablo Ortega This paper proves a version for stochastic differential equations of the Lie-Scheffers Theorem. This result characterizes the existence of nonlinear superposition rules for the general solution of those equations in terms of the involution properties of the distribution generated by the vector fields that define it. When stated in the particular case of standard deterministic systems, our main theorem improves various aspects of the classical Lie-Scheffers result. We show that the stochastic analog of the classical Lie-Scheffers systems can be reduced to the study of Lie group valued stochastic Lie-Scheffers systems; those systems, as well as those taking values in homogeneous spaces are studied in detail. The developments of the paper are illustrated with several examples. http://arxiv.org/abs/0803.0600 --------------------------------------------------------------- 6760. POLLING SYSTEMS WITH PARAMETER REGENERATION, THE GENERAL CASE Iain MacPhee and Mikhail Menshikov and Dimitri Petritis and Serguei Popov We consider a polling model with multiple stations, each with Poisson arrivals and a queue of infinite capacity. The service regime is exhaustive and there is Jacksonian feedback of served customers. What is new here is that when the server comes to a station it chooses the service rate and the feedback parameters at random; these remain valid during the whole stay of the server at that station. We give criteria for recurrence, transience, and existence of the $s$th moment of the return time to the empty state for this model. This paper generalizes the model when only two stations accept arriving jobs which was considered in \cite{MMPP}. Our results are stated in terms of Lyapunov exponents for random matrices. From the recurrence criteria it can be seen that the polling model with parameter regeneration can exhibit the unusual phenomenon of null recurrence over a thick region of parameter space. http://arxiv.org/abs/0803.0625 --------------------------------------------------------------- 6761. THE THEORY OF FALLIBLE PROBABILITY AND THE DYNAMICS OF DEGREES OF BELIEF Amos Nathan This monograph is an account of the theory of fallible probability and of the dynamics of degrees of belief. It discusses the first order subjective theory in which first order degrees of belief are expressed by subjective probabilities and are updated by conditionalization (Bayes, 1764; Ramsey, 1926), gives an improved exposition of the greater part of the author's theory of Probability Dynamics (Nathan, 2006) which should replace the so- called Probability Kinematics (Jeffrey, 1965), resolves the problem of New Explanation of Old Evidence (Jeffrey, 1995), provides a Theory of Confirmation, and refutes the Principle of Reflection (Van Fraassen, 1984). http://arxiv.org/abs/0803.0630 --------------------------------------------------------------- 6762. A NOTE ON OPTIMAL PROBABILITY LOWER BOUNDS FOR CENTERED RANDOM VARIABLES Mark Veraar In this note we obtain lower bounds for $\P(\xi\geq 0)$ and $ \P(\xi>0)$ under assumptions on the moments of a centered random variable $\xi$. The obtained estimates are shown to be optimal and improve results from the literature. The results are applied to obtain probability lower bounds for second order Rademacher chaos. http://arxiv.org/abs/0803.0727 --------------------------------------------------------------- 6763. RECURRENCE OF THE TWISTED PLANAR RANDOM WALK U. Haboeck We show that the "twisted" planar random walk - which results by summing up stationary increments rotated by multiples of a fixed angle - is recurrent under diverse assumptions on the increment process. For example, if the increment process is alpha-mixing and of finite second moment, then the twisted random walk is recurrent for every angle fixed choice of the angle out of a set of full Lebesgue measure, no matter how slow the mixing coefficients decay. http://arxiv.org/abs/0803.0724 --------------------------------------------------------------- 6764. TIME--SPACE HARMONIC POLYNOMIALS RELATIVE TO A L\'{E}VY PROCESS Josep Llu\'is Sol\'e and Frederic Utzet In this work, we give a closed form and a recurrence relation for a family of time--space harmonic polynomials relative to a L\'{e}vy process. We also state the relationship with the Kailath--Segall (orthogonal) polynomials associated to the process. http://arxiv.org/abs/0803.0829 --------------------------------------------------------------- 6765. ON THE RUIN PROBLEM IN THE RENEWAL RISK PROCESSES PERTURBED BY DIFFUSION Min Song In this paper, we consider the perturbed renewal risk process. Systems of integro-differential equations for the Gerber-Shiu functions at ruin caused by a claim and oscillation are established, respectively. The explicit Laplase transforms of Gerber-Shiu functions are obtained, while the closed form expressions for the Gerber-Shiu functions are derived when the claim amount distribution is from the rational family. Finally, we present numerical examples intended to illustrate the main results. http://arxiv.org/abs/0803.0906 --------------------------------------------------------------- 6766. ON THE SECOND-ORDER CORRELATION FUNCTION OF THE CHARACTERISTIC POLYNOMIAL OF A HERMITIAN WIGNER MATRIX F. G\"otze and H. K\"osters We consider the asymptotics of the second-order correlation function of the characteristic polynomial of a random matrix. We show that the known result for a random matrix from the Gaussian Unitary Ensemble essentially continues to hold for a general Hermitian Wigner matrix. Our proofs rely on an explicit formula for the exponential generating function of the second-order correlation function of the characteristic polynomial. http://arxiv.org/abs/0803.0926 --------------------------------------------------------------- 6767. ON THE SECOND-ORDER CORRELATION FUNCTION OF THE CHARACTERISTIC POLYNOMIAL OF A REAL SYMMETRIC WIGNER MATRIX H. K\"osters We consider the asymptotic behaviour of the second-order correlation function of the characteristic polynomial of a real symmetric random matrix. Our main result is that the existing result for a random matrix from the Gaussian Orthogonal Ensemble essentially continues to hold for a general real symmetric Wigner matrix. http://arxiv.org/abs/0803.0932 --------------------------------------------------------------- 6768. STATISTICAL ANALYSIS OF SELF-SIMILAR CONSERVATIVE FRAGMENTATION CHAINS Marc Hoffmann (LAMA) and Nathalie Krell (PMA) We explore statistical inference in self-similar conservative fragmentation chains, when only (approximate) observations of the size of the fragments below a given threshold are available. This framework, introduced by Bertoin and Martinez, is motivated by mineral crushing in mining industry. The underlying estimated object is the step distribution of the random walk associated to a randomly tagged fragment that evolves along the genealogical tree representation of the fragmentation process. We compute upper and lower rates of estimation in a parametric framework, and show that in the non- parametric case, the difficulty of the estimation is comparable to ill-posed linear inverse problems of order 1 in signal denoising. http://arxiv.org/abs/0803.0879 --------------------------------------------------------------- 6769. TOEPLITZ BLOCK MATRICES IN COMPRESSED SENSING Florian Sebert and Leslie Ying and and Yi Ming Zou Recent work in compressed sensing theory shows that $n\times N$ independent and identically distributed (IID) sensing matrices whose entries are drawn independently from certain probability distributions guarantee exact recovery of a sparse signal with high probability even if $n\ll N$. Motivated by signal processing applications, random filtering with Toeplitz sensing matrices whose elements are drawn from the same distributions were considered and shown to also be sufficient to recover a sparse signal from reduced samples exactly with high probability. This paper considers Toeplitz block matrices as sensing matrices. They naturally arise in multichannel and multidimensional filtering applications and include Toeplitz matrices as special cases. It is shown that the probability of exact reconstruction is also high. Their performance is validated using simulations. http://arxiv.org/abs/0803.0755 --------------------------------------------------------------- 6770. RANDOM MOTION WITH GAMMA-DISTRIBUTED ALTERNATING VELOCITIES IN BIOLOGICAL MODELING Antonio Di Crescenzo and Barbara Martinucci Motivated by applications in mathematical biology concerning randomly alternating motion of micro-organisms, we analyze a generalized integrated telegraph process. The random times between consecutive velocity reversals are gamma-distributed, and perform an alternating renewal process. We obtain the probability law and the mean of the process. http://arxiv.org/abs/0803.1067 --------------------------------------------------------------- 6771. PLANE RECURSIVE TREES, STIRLING PERMUTATIONS AND AN URN MODEL Svante Janson We exploit a bijection between plane recursive trees and Stirling permutations; this yields the equivalence of some results previously proven separately by different methods for the two types of objects as well as some new results. We also prove results on the joint distribution of the numbers of ascents, descents and plateaux in a random Stirling permutation. The proof uses an interesting generalized Polya urn http://arxiv.org/abs/0803.1129 --------------------------------------------------------------- 6772. LONG TIME BEHAVIOUR OF A STOCHASTIC-LAGRANGIAN PARTICLE SYSTEM FOR THE NAVIER-STOKES EQUATIONS Gautam Iyer and Jonathan Mattingly This paper is based on a formulation of the Navier-Stokes equations developed in \texttt{arxiv:math.PR/0511067} (to appear in Commun. Pure Appl. Math), where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. In this paper, we take $N$ copies of the above process (each based on independent Wiener processes), and replace the expected value with $\frac{1}{N}$ times the sum over these $N$ copies. We prove that in two dimensions, this system has (time) global solutions with $\holderspace{1}{\alpha}$ initial data. Further, we show that as $N \to \infty$ the system converges to the solution of Navier-Stokes equations on any finite interval $[0,T]$. However for fixed $N$, we prove that this system retains roughly $O(\frac{1}{N})$ times it's original energy $t \to \infty$. For general flows, we only provide a lower bound to this effect. In the special case of shear flows, we compute the behaviour as $t \to \infty$ explicitly. http://arxiv.org/abs/0803.1222 --------------------------------------------------------------- 6773. PARAMETER COLLAPSE DUE TO THE ZEROS IN THE INVERSE CONDITION R. Spjut Helton, Lasserre, and Putinar (2008, Ann. Probability; arXiv:math/ 0702314) expose the relationship between three properties of a measure: the conditional triangularity property of the associated orthogonal polynomials, the zeros in the inverse condition of the truncated moment matrix, and conditional independence. The purpose of this article is to provide examples of parameter collapse to product structure given that the zeros in the inverse condition holds up to some degree d. Specifically, start with a parameterized family of probability density functions; require that the zeros in the inverse condition up to degree d holds; and validate that imposing this restriction on the parameterized family results in a measure with product structure, or at least that conditional independence holds. Algorithms related to parameter collapse are supplied, including the computation of the zeros in the inverse condition up to degree d. http://arxiv.org/abs/0803.1225 --------------------------------------------------------------- 6774. ON SOME TRANSFORMATIONS OF BILATERAL BIRTH-AND-DEATH PROCESSES WITH APPLICATIONS Antonio Di Crescenzo A method yielding simple relationships among bilateral birth-and-death processes is outlined. This allows one to relate birth and death rates of two processes in such a way that their transition probabilities, first- passage-time densities and ultimate crossing probabilities are mutually related by some product-form expressions. http://arxiv.org/abs/0803.1413 --------------------------------------------------------------- 6775. ESTIMATION OF WIENER--ITO INTEGRALS AND POLYNOMIALS OF INDEPENDENT GAUSSIAN RANDOM VARIABLES Peter Major In this paper I prove good estimates on the moments and tail distribution of $k$-fold Wiener--It\^o integrals and also present their natural counterpart for polynomials of independent Gaussian random variables. The proof is based on the so-called diagram formula for Wiener--It\^o integrals which yields a good representation for their products as a sum of such integrals. I intend to show in a subsequent paper that this method also yields good estimates for degenerate $U$-statistics. The main result of this paper is a generalization of the estimates of Hanson and Wright about bilinear forms of independent standard normal random variables. On the other hand, it is a weaker estimate than the main result of a paper of Lata{\l}a [6]. But that paper contains an error, and it is not clear whether its result is true. This question is also discussed here. http://arxiv.org/abs/0803.1453 --------------------------------------------------------------- 6776. LYAPUNOV EXPONENTS FOR THE ONE-DIMENSIONAL PARABOLIC ANDERSON MODEL WITH DRIFT Alexander Drewitz We consider the solution $u$ to the one-dimensional parabolic Anderson model with homogeneous initial condition $u(0, \cdot) \equiv 1$, arbitrary drift and a time-independent potential bounded from above. Under ergodicity and independence conditions we derive representations for both the quenched Lyapunov exponent and, more importantly, the $p$-th annealed Lyapunov exponents for {\it all} $p \in (0, \infty).$ These results enable us to prove the heuristically plausible fact that the $p$-th annealed Lyapunov exponent converges to the quenched Lyapunov exponent as $p \downarrow 0.$ Furthermore, we show that $u$ is $p$-intermittent for $p$ large enough. As a byproduct, we compute the optimal quenched speed of the random walk appearing in the Feynman-Kac representation of $u$ under the corresponding Gibbs measure. In this context, depending on the negativity of the potential, a phase transition from zero speed to positive speed appears. http://arxiv.org/abs/0803.1480 --------------------------------------------------------------- 6777. DIFFRACTION OF STOCHASTIC POINT SETS: EXACTLY SOLVABLE EXAMPLES Michael Baake (Bielefeld) and Matthias Birkner (Berlin) and Robert V. Moody (Victoria) Stochastic point sets are considered that display a diffraction spectrum of mixed type, with special emphasis on explicitly computable cases together with a unified approach of reasonable generality. Several pairs of autocorrelation and diffraction measures are discussed which show a duality structure that may be viewed as analogues of the Poisson summation formula for lattice Dirac combs. http://arxiv.org/abs/0803.1266 --------------------------------------------------------------- 6778. REVERSIBILITY OF INTERACTING FLEMING-VIOT PROCESSES WITH MUTATION, SELECTION, AND RECOMBINATION Shui Feng and Byron Schmuland and Jean Vaillancourt and and Xiaowen Zhou Reversibility of the Fleming-Viot process with mutation, selection, and recombination is well understood. In this paper, we study the reversibility of a system of Fleming-Viot processes that live on a countable number of colonies interacting with each other through migrations between the colonies. It is shown that reversibility fails when both migration and mutation are non-trivial. http://arxiv.org/abs/0803.1492 --------------------------------------------------------------- 6779. ON SOME GENERALIZED REINFORCED RANDOM WALKS ON INTEGERS Olivier Raimond (LM-Orsay) and Bruno Schapira (LM-Orsay) We consider Reinforced Random Walks where transition probabilities are a function of the proportion of times the walk has traversed an edge. We give conditions for recurrence or transience. A phase transition is observed, similar to Pemantle \cite{Pem000} on trees. http://arxiv.org/abs/0803.1590 --------------------------------------------------------------- 6780. EQUALITY OF PRESSURES FOR DIFFEOMORPHISMS PRESERVING HYPERBOLIC MEASURES Katrin Gelfert For a diffeomorphism which preserves a hyperbolic measure the potential $\phi^u=-\log|{\rm Jac} df|_{E^u}|$ is studied. Various types of pressure of $\phi^u$ are introduced. It is shown that these pressures satisfy a corresponding variational principle. http://arxiv.org/abs/0803.1525 --------------------------------------------------------------- 6781. A NOTE ON MULTI-TYPE COOKIE RANDOM WALK ON INTEGERS Bruno Schapira (LM-Orsay) We consider a random walk on integers where at the first visits to a site the walker gets a positive drift, but where after a certain number of visits the walker gets a negative drift. We prove that the walker is almost surely transient to the left with positive speed. This is a variant of a model studied by Zerner, Kosygina and Zerner, and Basdevant and Singh. http://arxiv.org/abs/0803.1664 --------------------------------------------------------------- 6782. COPOLYMERS AT SELECTIVE INTERFACES: NEW BOUNDS ON THE PHASE DIAGRAM T. Bodineau and G. Giacomin and H. Lacoin and F. Toninelli We investigate the phase diagram of disordered copolymers at the interface between two selective solvents, and in particular its weak-coupling behavior, encoded in the slope $m_c$ of the critical line at the origin. In mathematical terms, the partition function of such a model does not depend on all the details of the Markov chain that models the polymer, but only on the time elapsed between successive returns to zero and on whether the walk is in the upper or lower half plane between such returns. This observation leads to a natural generalization of the model, in terms of arbitrary laws of return times: the most interesting case being the one of return times with power law tails (with exponent $1+\ga$, $\ga=1/2$ in the case of the symmetric random walk). The main results we present here are: 1/ The improvement of the known result $1/(1+\alpha)\le m_c\le 1$, as soon as $\ga >1$ for what concerns the upper bound, and down to $\ga\approx 0.65$ for the lower bound. 2/ A proof of the fact that the critical curve lies strictly below the critical curve of the annealed model for every non-zero value of the coupling parameter. We also provide an argument that rigorously shows the strong dependence of the phase diagram on the details of the return probability (and not only on the tail behavior). Lower bounds are obtained by exhibiting a new localization strategy, while upper bounds are based on estimates of non-integer moments of the partition function. http://arxiv.org/abs/0803.1766 --------------------------------------------------------------- 6783. BSDES WITH TWO RCLL REFLECTING OBSTACLES DRIVEN BY A BROWNIAN MOTION AND POISSON MEASURE AND RELATED MIXED ZERO-SUM GAMES S.Hamad\'ene and H.Wang In this paper we study Backward Stochastic Differential Equations with two reflecting right continuous with left limits obstacles (or barriers) when the noise is given by Brownian motion and a Poisson random measure mutually independent. The jumps of the obstacle processes could be either predictable or inaccessible. We show existence and uniqueness of the solution when the barriers are completely separated and the generator uniformly Lipschitz. We do not assume the existence of a difference of supermartingales between the obstacles. As an application, we show that the related mixed zero-sum differential-integral game problem has a value. http://arxiv.org/abs/0803.1815 --------------------------------------------------------------- 6784. BALANCE, GROWTH AND DIVERSITY OF FINANCIAL MARKETS Constantinos Kardaras A financial market comprising of a certain number of distinct companies is considered, and the following statement is proved: either a specific agent will surely beat the whole market unconditionally in the long run, or (and this "or" is not exclusive) all the capital of the market will accumulate in one company. Thus, absence of any "free unbounded lunches relative to the total capital" opportunities lead to the most dramatic failure of diversity in the market: one company takes over all other until the end of time. In order to prove this, we introduce the notion of perfectly balanced markets, which is an equilibrium state in which the relative capitalization of each company is a martingale under the physical probability. Then, the weaker notion of balanced markets is discussed where the martingale property of the relative capitalizations holds only approximately, we show how these concepts relate to growth- optimality and efficiency of the market, as well as how we can infer a shadow interest rate that is implied in the economy in the absence of a bank. http://arxiv.org/abs/0803.1858 --------------------------------------------------------------- 6785. THE NUMERAIRE PORTFOLIO IN SEMIMARTINGALE FINANCIAL MODELS Ioannis Karatzas and Constantinos Kardaras We study the existence of the numeraire portfolio under predictable convex constraints in a general semimartingale model of a financial market. The numeraire portfolio generates a wealth process, with respect to which the relative wealth processes of all other portfolios are supermartingales. Necessary and sufficient conditions for the existence of the numeraire portfolio are obtained in terms of the triplet of predictable characteristics of the asset price process. This characterization is then used to obtain further necessary and sufficient conditions, in terms of a no-free- lunch-type notion. In particular, the full strength of the "No Free Lunch with Vanishing Risk" (NFLVR) is not needed, only the weaker "No Unbounded Profit with Bounded Risk" (NUPBR) condition that involves the boundedness in probability of the terminal values of wealth processes. We show that this notion is the minimal a-priori assumption required in order to proceed with utility optimization. The fact that it is expressed entirely in terms of predictable characteristics makes it easy to check, something that the stronger NFLVR condition lacks. http://arxiv.org/abs/0803.1877 --------------------------------------------------------------- 6786. MONOTONICITY FOR EXCITED RANDOM WALK IN HIGH DIMENSIONS Remco van der Hofstad and Mark Holmes We prove that the drift $\theta(d,\beta)$ for excited random walk in dimension $d$ is monotone in the excitement parameter $\beta \in[0, 1]$, when $d\ge 9$. http://arxiv.org/abs/0803.1881 --------------------------------------------------------------- 6787. ON FINANCIAL MARKETS WHERE ONLY BUY-AND-HOLD TRADING IS POSSIBLE Constantinos Kardaras and Eckhard Platen A financial market model where agents can only trade using realistic buy-and-hold trategies is considered. Minimal assumptions are made on the nature of the asset-price process - in particular, the semimartingale property is not assumed. Via a natural assumption of limited opportunities for unlimited resulting wealth from trading, coined the No-Unbounded-Profit-with- Bounded-Risk (NUPBR) condition, we establish that asset-prices have be semimartingales, as well as a weakened version of the Fundamental Theorem of Asset Pricing that involves supermartingale deflators rather than equivalent martingale measures. Further, the utility maximization problem is considered and it is shown that using only buy-and-hold strategies, optimal utilities and wealth processes resulting from continuous trading can be approximated arbitrarily well. http://arxiv.org/abs/0803.1890 --------------------------------------------------------------- 6788. CONSTANTS OF CONCENTRATION FOR A SIMPLE RECURRENT RANDOM WALK ON RANDOM ENVIRONMENT Pierre Andreoletti (MAPMO) We precise the asymptotic of the limsup of the size of the neighborhood of concentration of Sinai's walk. Also we get the almost sure limits of the number of points visited more than a fixed proportion of a given amount of time. http://arxiv.org/abs/0803.2006 --------------------------------------------------------------- 6789. STRONG LAW OF LARGE NUMBERS WITH CONCAVE MOMENTS Anders Karlsson and Nicolas Monod In this note not intended for publication, it is observed that a wellnigh trivial application of the ergodic theorem of Karlsson-Ledrappier yields a strong LLN for arbitrary concave moments. http://arxiv.org/abs/0803.1856 --------------------------------------------------------------- 6790. DIFFUSION AT THE RANDOM MATRIX HARD EDGE Jose A. Ramirez and Brian Rider We show that the limiting minimal eigenvalue distributions for a natural generalization of Gaussian sample-covariance structures (the "beta ensembles") are described in by the spectrum of a random diffusion generator. By a Riccati transformation, we obtain a second diffusion description of the limiting eigenvalues in terms of hitting laws. This picture pertains to the so- called hard edge of random matrix theory and sits in complement to the recent work of the authors and B. Virag on the general beta random matrix soft edge. In fact, the diffusion descriptions found on both sides are used here to prove there exists a transition between the soft and hard edge laws at all values of beta. http://arxiv.org/abs/0803.2043 --------------------------------------------------------------- 6791. OPTIMAL TWO-VALUE ZERO-MEAN DISINTEGRATION OF ZERO-MEAN RANDOM VARIABLES Iosif Pinelis For any continuous zero-mean random variable (r.v.) X, a reciprocating function r is constructed, based only on the distribution of X, such that the conditional distribution of X given the (at-most-)two-point set {X,r(X)} is the zero-mean distribution on this set; in fact, a more general construction without the continuity assumption is given in this paper, as well as a large variety of other related results, including characterizations of the reciprocating function and modeling distribution asymmetry patterns. The mentioned disintegration of zero-mean r.v.'s implies, in particular, that an arbitrary zero-mean distribution is represented as the mixture of two- point zero-mean distributions; moreover, this mixture representation is most symmetric in a variety of senses. Somewhat similar representations -- of any probability distribution as the mixture of two-point distributions with the same skewness coefficient (but possibly with different means) -- go back to Kolmogorov; very recently, Aizenman et al. further developed such representations and applied them to (anti-)concentration inequalities for functions of independent random variables and to spectral localization for random Schroedinger operators. One kind of application given in the present paper is to construct certain statistical tests for asymmetry patterns and for location without symmetry conditions. Exact inequalities implying conservative properties of such tests are presented. These developments extend results established earlier by Efron, Eaton, and Pinelis under a symmetry condition. http://arxiv.org/abs/0803.2068 --------------------------------------------------------------- 6792. NO-FREE-LUNCH EQUIVALENCES FOR EXPONENTIAL LEVY MODELS Constantinos Kardaras We provide equivalence of numerous no-free-lunch type conditions for financial markets where the asset prices are modeled as exponential Levy processes, under possible convex constraints in the use of investment strategies. The general message is the following: if any kind of free lunch exists in these models it has to be of the most egregious type, generating an increasing ealth. Furthermore, we connect the previous to the existence of the numeraire portfolio, both for its particular expositional clarity in exponential Levy models and as a first step in obtaining analogues of the no-free-lunch equivalences in general semimartingale models. http://arxiv.org/abs/0803.2169 --------------------------------------------------------------- 6793. ON AGENTS' AGREEMENT AND PARTIAL-EQUILIBRIUM PRICING IN INCOMPLETE MARKETS Michail Anthropelos and Gordan Zitkovic We consider two risk-averse financial agents who negotiate the price of an illiquid indivisible contingent claim in an incomplete semimartingale market environment. Under the assumption that the agents are exponential utility maximizers with non-traded random endowments, we provide necessary and sufficient conditions for negotiation to be successful, i.e., for the trade to occur. We also study the asymptotic case where the size of the claim is small compared to the random endowments and we give a full characterization in this case. Finally, we study a partial-equilibrium problem for a bundle of divisible claims and establish existence and uniqueness. A number of technical results on conditional indifference prices are provided. http://arxiv.org/abs/0803.2198 --------------------------------------------------------------- 6794. DUALITY OF CHORDAL SLE, II Dapeng Zhan We improve the geometric properties of SLE$(\kappa;\vec{\rho})$ processes derived in an earlier paper, which are then used to obtain more results about the duality of SLE. We find that for $\kappa\in (4,8)$, the boundary of a standard chordal SLE$(\kappa)$ hull stopped on swallowing a fixed $x\in\R\sem\{0\}$ is the image of some SLE$(16/\kappa;\vec{\rho})$ trace started from a random point. Using this fact together with a similar proposition in the case that $\kappa\ge 8$, we obtain a description of the boundary of a standard chordal SLE$(\kappa)$ hull for $\kappa>4$, at a finite stopping time. Finally, we prove that for $\kappa>4$, in many cases, the limit of a chordal or strip SLE$(\kappa;\vec{\rho})$ trace exists. http://arxiv.org/abs/0803.2223 --------------------------------------------------------------- 6795. A DECOMPOSITION OF THE BIFRACTIONAL BROWNIAN MOTION AND SOME APPLICATIONS Pedro Lei and David Nualart In this paper we show a decomposition of the bifractional Brownian motion with parameters H,K into the sum of a fractional Brownian motion with Hurst parameter HK plus a stochastic process with absolutely continuous trajectories. Some applications of this decomposition are discussed. http://arxiv.org/abs/0803.2227 --------------------------------------------------------------- 6796. RANDOM SOLUTIONS OF RANDOM PROBLEMS...ARE NOT JUST RANDOM Dimitris Achlioptas and Amin Coja-Oghlan Let I(n,m) denote a uniformly random instance of some constraint satisfaction problem CSP with n variables and m constraints. Assume that the density r=m/n is small enough so that with high probability I(n,m) has a solution, and consider the experiment of first choosing an instance I=I(n,m) at random, and then sampling a random solution sigma of I (if one exists). For many CSPs (e.g., k-SAT, k-NAE, or k-coloring), this experiment appears difficult both to implement and to analyze; in fact, for a large range of r, no efficient algorithm is known to even compute a single solution of I. In the present paper we show that for many CSPs the above experiment is essentially equivalent to first choosing a random assignment sigma to the n variables, and then drawing a random instance satisfied by sigma uniformly. In general, this second experiment is very easy to implement and amenable to a rigorous analysis. In fact, using this equivalence, we can analyze the solution space of random CSPs. Thus, we can achieve the long-standing goal of establishing rigorously a picture put forward by statistical physicists on the basis of sophisticated but non-rigorous techniques such as the cavity and the replica method. This picture is suggestive as to why random CSP instances seem difficult to deal with algorithmically. Furthermore, we show that the second experiment gives rise to one-way functions, if one assumes that random instances of CSP are hard for some range of densities. http://arxiv.org/abs/0803.2122 --------------------------------------------------------------- 6797. PARTIAL DIFFERENTIAL EQUATIONS DRIVEN BY ROUGH PATHS Michael Caruana and Peter Friz We study a class of linear first and second order partial differential equations driven by weak geometric $p$-rough paths, and prove the existence of a unique solution for these equations. This solution depends continuously on the driving rough path. This allows a robust approach to stochastic partial differential equations. In particular, we may replace Brownian motion by more general Gaussian and Markovian noise. Support theorems and large deviation statements all became easy corollaries of the corresponding statements of the driving process. In the case of first order equations with Gaussian noise, we discuss the existence of a density with respect to the Lebesgue measure for the solution. http://arxiv.org/abs/0803.2178 --------------------------------------------------------------- 6798. ON PERPETUAL AMERICAN PUT VALUATION AND FIRST-PASSAGE IN A REGIME-SWITCHING MODEL WITH JUMPS Z. Jiang and M.R. Pistorius In this paper we consider the problem of pricing a perpetual American put option in an exponential regime-switching L\'{e}vy model. For the case of the (dense) class of phase-type jumps and finitely many regimes we derive an explicit expression for the value function. The solution of the corresponding first passage problem under a state-dependent level rests on a path transformation and a new matrix Wiener-Hopf factorization result for this class of processes. http://arxiv.org/abs/0803.2302 --------------------------------------------------------------- 6799. ABSOLUTE CONTINUITY AND CONVERGENCE IN VARIATION FOR DISTRIBUTIONS OF A FUNCTIONALS OF POISSON POINT MEASURE Alexey M.Kulik General sufficient conditions are given for absolute continuity and convergence in variation of distributions of a functionals on a probability space, generated by a Poisson point measure. The phase space of the Poisson point measure is supposed to be of the form (0,\infty)\times U, and its intensity measure to be equal dt\Pi(du). We introduce the family of time stretching transformations of the configurations of the point measure. The sufficient conditions for absolute continuity and convergence in variation are given in the terms of the time stretching transformations and the relative differential operators. These conditions are applied to solutions of SDE's driven by Poisson point measures, including an SDE's with non-constant jump rate. http://arxiv.org/abs/0803.2389 --------------------------------------------------------------- 6800. ON A SZEGO TYPE LIMIT THEOREM, THE HOLDER-YOUNG-BRASCAMP-LIEB INEQUALITY, AND THE ASYMPTOTIC THEORY OF INTEGRALS AND QUADRATIC FORMS OF STATIONARY FIELDS Florin Avram (LMA-PAU) and Nikolai Leonenko and Ludmila Sakhno Many statistical applications require establishing central limit theorems for sums, integrals, or for quadratic forms of functions of a stationary process. A particularly important case is that of Appell polynomials, since the Appell expansion rank" determines typically the type of central limit theorem satisfied by these functionals. We review and extend here to multidimensional indices a functional analysis approach to this problem proposed by Avram and Brown (1989), based on the method of cumulants and on integrability assumptions in the spectral domain; several applications are presented as well. http://arxiv.org/abs/0803.2441 --------------------------------------------------------------- 6801. STABILITY OF A PROCESSOR SHARING QUEUE WITH VARYING THROUGHPUT Pascal Moyal In this paper, we present a stability criterion for Processor Sharing queues, in which the throughput may depend on the number of customers in the system (in such cases such as interferences between the users). Such a system is represented by a point measure-valued stochastic recursion keeping track of the remaining processing times of the customers. http://arxiv.org/abs/0803.2459 --------------------------------------------------------------- 6802. ON THE LEAST SQUARES ESTIMATOR IN A NEARLY UNSTABLE SEQUENCE OF STATIONARY SPATIAL AR MODELS S\'andor Baran and Gyula Pap A nearly unstable sequence of stationary spatial autoregressive processes is investigated, when the sum of the absolute values of the autoregressive coefficients tends to one. It is shown that after an appropriate norming the least squares estimator for these coefficients has a normal limit distribution. If none of the parameters equals zero than the typical rate of convergence is n. http://arxiv.org/abs/0803.2486 --------------------------------------------------------------- 6803. A RUELLE OPERATOR FOR CONTINUOUS TIME MARKOV CHAINS Alexandre Baraviera and Ruy Exel and Artur O. Lopes We consider a generalization of the Ruelle theorem for the case of continuous time problems. We present a result which we believe is important for future use in problems in Mathematical Physics related to $C^*$-Algebras We consider a finite state set $S$ and a stationary continuous time Markov Chain $X_t $, $t\geq 0$, taking values on $S$. We denote by $\Omega$ the set of paths $w$ taking values on $S$ (the elements $w$ are locally constant with left and right limits and are also right continuous on $t$). We consider an infinitesimal generator $L$ and a stationary vector $p_0$. We denote by $P$ the associated probability on ($\Omega, {\cal B}$). All functions $f$ we consider bellow are in the set ${\cal L}^\infty (P)$. From the probability $P$ we define a Ruelle operator ${\cal L}^t, t\geq 0$, acting on functions $f:\Omega \to \mathbb{R}$ of ${\cal L}^\infty (P)$. Given $V:\Omega \to \mathbb{R}$, such that is constant in sets of the form $\{X_0=c\}$, we define a modified Ruelle operator $\tilde{{\cal L}}_V^t, t\geq 0$. We are able to show the existence of an eigenfunction $u$ and an eigen-probability $\nu_V$ on $\Omega$ associated to $\tilde{{\cal L}}^t_V, t\geq 0$. We also show the following property for the probability $\nu_V$: for any integrable $g\in {\cal L}^\infty (P)$ and any real and positive $t$ $$ \int e^{-\int_0^t (V \circ \Theta_s)(.) ds} [ (\tilde{{\cal L}}^t_V (g)) \circ \theta_t ] d \nu_V = \int g d \nu_V$$ This equation generalize, for the continuous time Markov Chain, a similar one for discrete time systems (and which is quite important for understanding the KMS states of certain $C^*$-algebras). http://arxiv.org/abs/0803.2501 --------------------------------------------------------------- 6804. A CIRCLE OF INTERACTING SERVERS; SPONTANEOUS COLLECTIVE BEHAVIOR IN CASE OF LARGE FLUCTUATIONS E.A. Pechersky and N.D. Vvedenskaya We consider large fluctuations, namely overload of servers, in a network with dynamic routing of messages. The servers form a circle. The number of input flows is equal to the number of servers, the messages of any flow are distributed between two neighboring servers, upon its arrival a message is directed to the least loaded of these servers. Under the condition that at least two servers are overloaded the number of overloaded servers in such network depends on the rate of input flows. In particular there exists critical level of input rate that in case of higher rate most probable that all servers are overloaded. http://arxiv.org/abs/0803.2576 --------------------------------------------------------------- 6805. MARKOV CHAINS APPROXIMATIONS OF JUMP-DIFFUSION QUANTUM TRAJECTORIES Clement Pellegrini (ICJ) "Quantum trajectories" are solutions of stochastic differential equations also called Belavkin or Stochastic Schr\"odinger Equations. They describe random phenomena in quantum measurement theory. Two types of such equations are usually considered, one is driven by a one-dimensional Brownian motion and the other is driven by a counting process. In this article, we present a way to obtain more advanced models which use jump-diffusion stochastic differential equations. Such models come from solutions of martingale problems for infinitesimal generators. These generators are obtained from the limit of generators of classical Markov chains which describe discrete models of quantum trajectories. Furthermore, stochastic models of jump-diffusion equations are physically justified by proving that their solutions can be obtained as the limit of the discrete trajectories. http://arxiv.org/abs/0803.2593 --------------------------------------------------------------- 6806. POISSON AND DIFFUSION APPROXIMATION OF STOCHASTIC SCHRODINGER EQUATIONS WITH CONTROL Clement Pellegrini (ICJ) "Quantum trajectories" are solutions of stochastic differential equations of non-usual type. Such equations are called ``Belavkin'' or ``Stochastic Schr\"odinger Equations'' and describe random phenomena in continuous measurement theory of Open Quantum System. Many recent investigations deal with the control theory in such model. In this article, stochastic models are mathematically and physically justified as limit of concrete discrete procedures called ``Quantum Repeated Measurements''. In particular, this gives a rigorous justification of the Poisson and diffusion approximation in quantum measurement theory with control. Furthermore we investigate some examples using control in quantum mechanics. http://arxiv.org/abs/0803.2643 --------------------------------------------------------------- 6807. A NEW CENTRAL LIMIT THEOREM UNDER SUBLINEAR EXPECTATIONS Shige Peng We describe a new framework of a sublinear expectation space and the related notions and results of distributions, independence. A new notion of G-distributions is introduced which generalizes our G-normal- distribution in the sense that mean-uncertainty can be also described. W present our new result of central limit theorem under sublinear expectation. This theorem can be also regarded as a generalization of the law of large number in the case of mean-uncertainty. http://arxiv.org/abs/0803.2656 --------------------------------------------------------------- 6808. FIELD THEORY CONJECTURE FOR LOOP-ERASED RANDOM WALKS Andrei A. Fedorenko and Pierre Le Doussal and Kay Joerg Wiese We give evidence that the functional renormalization group (FRG), developed to study disordered systems, may provide a field theoretic description for the loop-erased random walk (LERW), allowing to compute its fractal dimension in a systematic expansion in epsilon=4-d. Up to two loop, the FRG agrees with rigorous bounds, correctly reproduces the leading logarithmic corrections at the upper critical dimension d=4, and compares well with numerical studies. We obtain the universal subleading logarithmic correction in d=4, which can be used as a further test of the conjecture. http://arxiv.org/abs/0803.2357 --------------------------------------------------------------- 6809. DISCRETE STOCHASTIC PROCESSES, REPLICATOR AND FOKKER-PLANCK EQUATIONS OF COEVOLUTIONARY DYNAMICS IN FINITE AND INFINITE POPULATIONS Jens Christian Claussen Finite-size fluctuations in coevolutionary dynamics arise in models of biological as well as of social and economic systems. This brief tutorial review surveys a systematic approach starting from a stochastic process discrete both in time and state. The limit $N\to \infty$ of an infinite population can be considered explicitly, generally leading to a replicator-type equation in zero order, and to a Fokker-Planck-type equation in first order in $1/\sqrt{N}$. Consequences and relations to some previous approaches are outlined. http://arxiv.org/abs/0803.2443 --------------------------------------------------------------- 6810. TYPICAL DISPERSION AND GENERALIZED LYAPUNOV EXPONENTS Steven Finch and Zai-Qiao Bai and Pascal Sebah Let f(n) denote the number of odd entries in the nth row of Pascal's binomial triangle. We study "average dispersion" and "typical dispersion" of f(n) -- the latter involves computing a generalized Lyapunov exponent -- and then turn to numerical analysis of higher dimensional examples. http://arxiv.org/abs/0803.2611 --------------------------------------------------------------- 6811. POTTS MODELS IN THE CONTINUUM. UNIQUENESS AND EXPONENTIAL DECAY IN THE RESTRICTED ENSEMBLES A. De Masi and I. Merola and E. Presutti and Y. Vignaud In this paper we study a continuum version of the Potts model. Particles are points in R^d, with a spin which may take S possible values, S being at least 3. Particles with different spins repel each other via a Kac pair potential. In mean field, for any inverse temperature there is a value of the chemical potential at which S+1 distinct phases coexist. For each mean field pure phase, we introduce a restricted ensemble which is defined so that the empirical particles densities are close to the mean field values. Then, in the spirit of the Dobrushin Shlosman theory, we get uniqueness and exponential decay of correlations when the range of the interaction is large enough. In a second paper, we will use such a result to implement the Pirogov-Sinai scheme proving coexistence of S+1 extremal DLR measures. http://arxiv.org/abs/0803.2767 --------------------------------------------------------------- 6812. ON MONGE-KANTOROVICH PROBLEM IN THE PLANE Yinfang Shen and Weian Zheng We transfer the celebrating Monge-Kontorovich problem in a bounded domain of Euclidean plane into a Dirichlet boundary problem associated to a quasi-linear elliptic equation with $0-$order term missing in its diffusion coefficients: \begin{eqnarray*} A(x, F'_x)F''_{xx}+B(y, F'_y)F''_{yy}&=&C(x, y, F'_x, F'_y) \end{eqnarray*} where $A(.,.)>0, B(.,.)>0$ and $C$ are functions based on the initial distributions, $F$ is an unknown probability distribution function and therefore closed the former problem. http://arxiv.org/abs/0803.2830 --------------------------------------------------------------- 6813. SELF-REPELLING RANDOM WALK WITH DIRECTED EDGES ON Z Balint Toth and Balint Veto We consider a variant of self-repelling random walk on the integer lattice Z where the self-repellence is defined in terms of the local time on oriented edges. The long-time asymptotic scaling of this walk is surprisingly different from the asymptotics of the similar process with self-repellence defined in terms of local time on unoriented edges. We prove limit theorems for the local time process and for the position of the random walker. The main ingredient is a Ray-Knight-type of approach. At the end of the paper, we also present some computer simulations which show the strange scaling behaviour of the walk considered. http://arxiv.org/abs/0803.2848 --------------------------------------------------------------- 6814. PRODUCT-FORM STATIONARY DISTRIBUTIONS FOR DEFICIENCY ZERO CHEMICAL REACTION NETWORKS David F. Anderson and Gheorghe Craciun and Thomas G. Kurtz We consider both deterministically and stochastically modeled chemical reaction systems and prove that a product-form stationary distribution exists for each closed, irreducible subset of the state space of a stochastically modeled system if the corresponding deterministically modeled system admits a complex balanced equilibrium. Feinberg's deficiency zero theorem then implies that such a distribution exists so long as the corresponding chemical network is weakly reversible and has a deficiency of zero. We also demonstrate that the main parameter of the stationary distribution for the stochastically modeled system is a complex balanced equilibrium value for the corresponding deterministically modeled system. http://arxiv.org/abs/0803.3042 --------------------------------------------------------------- 6815. REPEATED QUANTUM INTERACTIONS QUANTUM LANGEVIN EQUATION AND THE LOW DENSITY LIMIT Ameur Dhahri (CEREMADE) We consider a repeated quantum interaction model describing a small system $\Hh_S$ in interaction with each one of the identical copies of the chain $\bigotimes_{\N^*}\C^{n+1}$, modeling a heat bath, one after another during the same short time intervals $[0,h]$. We suppose that the repeated quantum interaction Hamiltonian is split in two parts: a free part and an interaction part with time scale of order $h$. After giving the GNS representation, we establish the relation between the time scale $h$ and the classical low density limit. We introduce a chemical potential $\mu$ related to the time $h$ as follows: $h^2=e^{\beta\mu}$. We further prove that the solution of the associated discrete evolution equation converges strongly, when $h$ tends to 0, to the unitary solution of a quantum Langevin equation directed by Poisson processes. http://arxiv.org/abs/0803.3059 --------------------------------------------------------------- 6816. A LINDBLAD MODEL FOR A SPIN CHAIN COUPLED TO HEAT BATHS Ameur Dhahri (ICJ and Ceremade) We study a XY model which consists of a spin chain coupled to heat baths. We give a repeated quantum interaction Hamiltonian describing this model. We compute the explicit form of the associated Lindblad generator in the case of the spin chain coupled to one, two and several heat baths. We further study the properties of quantum master equation such as approach to equilibrium, local equilibrium states, entropy production and quantum detailed balance condition. http://arxiv.org/abs/0803.3060 --------------------------------------------------------------- 6817. RECORDS IN A CHANGING WORLD Joachim Krug In the context of this paper, a record is an entry in a sequence of random variables (RV's) that is larger or smaller than all previous entries. After a brief review of the classic theory of records, which is largely restricted to sequences of independent and identically distributed (i.i.d.) RV's, new results for sequences of independent RV's with distributions that broaden or sharpen with time are presented. In particular, we show that when the width of the distribution grows as a power law in time $n$, the mean number of records is asymptotically of order $\ln n$ for distributions with a power law tail (the \textit{Fr\'echet class} of extremal value statistics), of order $(\ln n)^2$ for distributions of exponential type (\textit{Gumbel class}), and of order $n^{1/(\nu+1)}$ for distributions of bounded support (\textit{Weibull class}), where the exponent $\nu$ describes the behaviour of the distribution at the upper (or lower) boundary. Simulations are presented which indicate that, in contrast to the i.i.d. case, the sequence of record breaking events is correlated in such a way that the variance of the number of records is asymptotically smaller than the mean. http://arxiv.org/abs/cond-mat/0702136 --------------------------------------------------------------- 6818. LOCAL SEMICIRCLE LAW AND COMPLETE DELOCALIZATION FOR WIGNER RANDOM MATRICES Laszlo Erdos and Benjamin Schlein and Horng-Tzer Yau We consider $N\times N$ Hermitian random matrices with i.i.d. entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. Under suitable assumptions on the distribution of the single matrix element, we prove that, away from the spectral edges, the density of eigenvalues concentrates around the Wigner semicircle law on energy scales $\eta \gg N^{-1} (\log N)^8$. Up to the logarithmic factor, this is the smallest energy scale for which the semicircle law may be valid. We also prove that for all eigenvalues away from the spectral edges, the $\ell^\infty $-norm of the corresponding eigenvectors is of order $O(N^{-1/2})$, modulo logarithmic corrections. The upper bound $O(N^{-1/2})$ implies that every eigenvector is completely de-localized, i.e., the maximum size of the components of the eigenvector is of the same order as their average size. In the Appendix, we include a lemma by J. Bourgain which removes one of our assumptions on the distribution of the matrix elements. http://arxiv.org/abs/0803.0542 --------------------------------------------------------------- 6819. INTERACTING PARTICLE SYSTEMS OUT OF EQUILIBRIUM Thomas Kriecherbauer and Joachim Krug These notes are based on lectures delivered by the authors at the Langeoog seminar of SFB/TR12 "Symmetries and universality in mesoscopic systems" in November 2007, to a mixed audience of mathematicians and theoretical physicists. After a brief outline of the basic physical concepts of equilibrium and nonequilibrium states, the one-dimensional totally asymmetric simple exclusion process (TASEP) is introduced as a paradigmatic nonequilibrium interacting particle system. The stationary measure on the ring is derived and the idea of the hydrodynamic limit is sketched. We then explain in detail a famous rigorous result due to Johansson, which relates the TASEP current fluctuations to the Tracy-Widom distribution of random matrix theory, and discuss its implications within the framework of the phenomenological Kardar-Parisi-Zhang equation. http://arxiv.org/abs/0803.2796 --------------------------------------------------------------- 6820. MINORS IN RANDOM REGULAR GRAPHS N. Fountoulakis and D. K\"uhn and D. Osthus We show that there is a constant c>0 so that for any fixed r which is at least 3 a.a.s. an r-regular graph on n vertices contains a complete graph on c n^{1/2} vertices as a minor. This confirms a conjecture of Markstrom. Since any minor of an r-regular graph on n vertices has at most rn/2 edges, our bound is clearly best possible up to the value of the constant c. As a corollary, we also obtain the likely order of magnitude of the largest complete minor in a random graph G(n,p) during the phase transition (i.e. when pn is close to 1). http://arxiv.org/abs/0803.3001 --------------------------------------------------------------- 6821. DIVERSITY AND RELATIVE ARBITRAGE IN EQUITY MARKETS Robert Fernholz and Ioannis Karatzas and Constantinos Kardaras A financial market is called "diverse" if no single stock is ever allowed to dominate the entire market in terms of relative capitalization. In the context of the standard Ito-process model initiated by Samuelson (1965) we formulate this property (and the allied, successively weaker notions of "weak diversity" and "asymptotic weak diversity") in precise terms. We show that diversity is possible to achieve, but delicate. Several illustrative examples are provided, which demonstrate that weakly-diverse financial markets contain relative arbitrage opportunities: it is possible to outperform (or underperform) such markets over sufficiently long time-horizons, and to underperform them significantly over arbitrary time-horizons. The existence of such relative arbitrage does not interfere with the development of option pricing, and has interesting consequences for the pricing of long-term warrants and for put-call parity. Several open questions are suggested for further study. http://arxiv.org/abs/0803.3093 --------------------------------------------------------------- 6822. REGENERATIVE TREE GROWTH: BINARY SELF-SIMILAR CONTINUUM RANDOM TREES AND POISSON-DIRICHLET COMPOSITIONS Jim Pitman and Matthias Winkel We use a natural ordered extension of the Chinese Restaurant Process to grow a two-parameter family of binary self-similar continuum fragmentation trees. We provide an explicit embedding of Ford's sequence of alpha model trees in the continuum tree which we identified in a previous article as a distributional scaling limit of Ford's trees. In general, the Markov branching trees induced by the two-parameter growth rule are not sampling consistent, so the existence of compact limiting trees cannot be deduced from previous work on the sampling consistent case. We develop here a new approach to establish such limits, based on regenerative interval partitions and the urn-model description of sampling from Dirichlet random distributions. http://arxiv.org/abs/0803.3098 --------------------------------------------------------------- 6823. A FEW REMARKS ON THE OPERATOR NORM OF RANDOM TOEPLITZ MATRICES Rados{\l}aw Adamczak We present some results concerning the almost sure behaviour of the operator norm or random Toeplitz matrices, including the law of large numbers for the norm, normalized by its expectation (in the i.i.d. case). As tools we present some concentration inequalities for suprema of empirical processes, which are refinements of recent results by Einmahl and Li. http://arxiv.org/abs/0803.3111 --------------------------------------------------------------- 6824. SYMMETRIC JUMP PROCESSES: LOCALIZATION, HEAT KERNELS, AND CONVERGENCE Richard F. Bass and Moritz Kassmann and and Takashi Kumagai We consider symmetric processes of pure jump type. We prove local estimates on the probability of exiting balls, the H\"older continuity of harmonic functions and of heat kernels, and convergence of a sequence of such processes. http://arxiv.org/abs/0803.3164 --------------------------------------------------------------- 6825. ON SOME RESULTS OF CUFARO PETRONI ABOUT STUDENT T-PROCESSES C. Berg and C. Vignat This paper deals with Student t-processes as studied in (Cufaro Petroni N 2007 J. Phys. A, Math. Theor. 40(10), 2227-2250). We prove and extend some conjectures expressed by Cufaro Petroni about the asymptotical behavior of a Student t-process and the expansion of its density. First, the explicit asymptotic behavior of any real positive convolution power of a Student t-density with any real positive degrees of freedom is given in the multivariate case; then the integer convolution power of a Student t-distribution with odd degrees of freedom is shown to be a convex combination of Student t-densities with odd degrees of freedom. At last, we show that this result does not extend to the case of non-integer convolution powers. http://arxiv.org/abs/0803.3198 --------------------------------------------------------------- 6826. RECURRENCE AND TRANSIENCE OF A MULTI-EXCITED RANDOM WALK ON A REGULAR TREE Anne-Laure Basdevant and Arvind Singh We study a model of multi-excited random walk on a regular tree which generalizes the models of the once excited random walk and the digging random walk introduced by Volkov (2003). We show the existence of a phase transition of the recurrence/transience property of the walk. In particular, we prove that the asymptotic behavior of the walk depends on the order of the excitations, which contrasts with the one dimensional setting studied by Zerner (2005). Special attention is given to the cases of the once excited, the twice excited and the digging random walk where explicit criterions, depending on the initial cookie environment, are provided to determine whether the walk is recurrent or transient. http://arxiv.org/abs/0803.3284 --------------------------------------------------------------- 6827. BRANCHING PROCESS APPROACH FOR 2-SAT THRESHOLDS Elchanan Mossel (UC Berkeley) and Arnab Sen (UC Berkeley) It is well known that, as $n$ tends to infinity, the probability of satisfiability for a random 2-SAT formula on $n$ variables, where each clause occurs independently with probability $\alpha/2n$, exhibits a sharp threshold at $\alpha=1$. We provide a simple conceptual proof of this fact based on branching process arguments. We also study a generalized 2-SAT model in which each clause occurs independently but with probability $\alpha_i/2n$ where $i \in \{0,1,2 \}$ is the number of positive literals in that clause. We use 2-type branching process arguments to determine the satisfiability threshold for this model in terms of the maximum eigenvalue of the branching matrix. http://arxiv.org/abs/0803.3285 --------------------------------------------------------------- 6828. A GENERALIZED FEYNMAN-KAC FORMULA FOR ONE DIMENSIONAL PROCESSES George Lowther Suppose that a real valued process X is given as a solution to a stochastic differential equation. Then, for any twice continuously differentiable function f, the Feynman-Kac formula gives a condition for f(t,X) to be a local martingale. We generalize the Feynman-Kac formula in two main ways. First, it is extended to nondifferentiable functions. Second, the process X is not required to satisfy an SDE. Instead, it is only required to be a quasimartingale satisfying an integrability condition, and the martingale condition for f(t,X) is then expressed in terms of the marginal distributions, drift measure and jumps of X. The proof involves the stochastic calculus of Dirichlet processes and a time-reversal argument. These results are then applied to show that a continuous and strong Markov martingale is uniquely determined by its marginal distributions. http://arxiv.org/abs/0803.3303 --------------------------------------------------------------- 6829. ASYMPTOTICS OF INPUT-CONSTRAINED BINARY SYMMETRIC CHANNEL CAPACITY Guangyue Han and Brian Marcus In this paper, we study the classical problem of noisy constrained capacity in the case of the binary symmetric channel (BSC), namely, the capacity of a BSC whose input is a sequence from a constrained set. Motivated by a result of Ordentlich and Weissman, we derive an asymptotic formula (when the noise parameter is small) for the entropy rate of a hidden Markov chain, observed when a Markov chain passes through a binary symmetric channel. Using this result we establish an asymptotic formula for the capacity of a binary symmetric channel with input process supported on an irreducible finite type constraint, as the noise parameter tends to zero. http://arxiv.org/abs/0803.3360 --------------------------------------------------------------- 6830. MULTIVARIATE ANALYSIS AND JACOBI ENSEMBLES: LARGEST EIGENVALUE, TRACY WIDOM LIMITS AND RATES OF CONVERGENCE Iain M. Johnstone Let $A$ and $B$ be independent, central Wishart matrices in $p$ variables with common covariance and having $m$ and $n$ degrees of freedom respectively. The distribution of the largest eigenvalue of $(A+B)^{-1}B$ has numerous applications in multivariate statistics, but is difficult to calculate exactly. Suppose that $m$ and $n$ grow in proportion to $p$. We show that after centering and scaling, the distribution is approximated to second order, $O(p^{-2/3})$, by the Tracy-Widom law. The results are obtained for both complex and then real valued data by using methods of random matrix theory to study the largest eigenvalue of the Jacobi unitary and orthogonal ensembles. Asymptotic approximations of Jacobi polynomials near the largest zero play a central role. http://arxiv.org/abs/0803.3408 --------------------------------------------------------------- 6831. HOMOGENIZATION FOR SEMI-LINEAR PDE WITH DISCONTINUOUS COEFFICIENTS K. Bahlali (IMATH) and Abouo Elouaflin (UFR-MI) and E. Pardoux (CMI) We study the asymptotic behavior of the solution of semi-linear PDEs. Neither periodicity nor ergodicity assumptions are assumed. The coefficients admit only a limit in a Cesaro sense. In such a case, the limit coefficients may have discontinuity. We use probabilistic approach based on weak convergence techniques for the associated backward stochastic differential equation in the S-topology. We establish weak continuity for the flow of the limit diffusion process and related the PDE limit to the backward stochastic differential equation via the representation of Lp-viscosity solution. http://arxiv.org/abs/0803.3499 --------------------------------------------------------------- 6832. MAXIMA OF DIRICHLET AND TRIANGULAR ARRAYS OF GAMMA VARIABLES Arup Bose and Amites Dasgupta and Krishanu Maulik Consider a rowwise independent triangular array of gamma random variables with varying parameters. Under several different conditions on the shape parameter, we show that the sequence of row-maximums converges weakly after linear or power transformation. Depending on the parameter combinations, we obtain both Gumbel and non-Gumbel limits. The weak limits for maximum of the coordinates of certain Dirichlet vectors of increasing dimension are also obtained using the gamma representation. http://arxiv.org/abs/0803.3518 --------------------------------------------------------------- 6833. INTEGRATION WITH RESPECT TO LOCAL TIME AND ITO'S FORMULA FOR SMOOTH NONDEGENERATE MARTINGALES Xavier Bardina and Carles Rovira We show an It\^ o's formula for nondegenerate Brownian martingales $X_t=\int_0^t u_s dW_s$ and functions $F(x,t)$ with locally integrable derivatives in $t$ and $x$. We prove that one can express the additional term in It\^o's s formula as an integral over space and time with respect to local time. http://arxiv.org/abs/0803.3522 --------------------------------------------------------------- 6834. ESCAPING THE BROWNIAN STALKERS Alexander Weiss We propose a simple model for the behaviour of longterm investors on a stock market, consisting of three particles, which represent the current price of the stock and the opinion of the buyers, respectively sellers, about the right trading price. As time evolves, both groups of traders update their opinions with respect to the current price. The update speed is controled by a parameter $\gamma$, the price process is described by a geometric Brownian motion. We consider the stability of the market in terms of the distance between the buyers' and sellers' opinion, and prove that the distance process is recurrent/transient in dependence on $\gamma$. http://arxiv.org/abs/0803.3590 --------------------------------------------------------------- 6835. ON CONVERGENCE OF DYNAMICS OF HOPPING PARTICLES TO A BIRTH-AND- DEATH PROCESS IN CONTINUUM Dmitri Finkelshtein and Yuri Kondratiev and Eugene Lytvynov We show that some classes of birth-and-death processes in continuum (Glauber dynamics) may be derived as a scaling limit of a dynamics of interacting hopping particles (Kawasaki dynamics) http://arxiv.org/abs/0803.3551 --------------------------------------------------------------- 6836. THE EQUIVALENCE BETWEEN UNIQUENESS AND CONTINUOUS DEPENDENCE OF SOLUTION FOR BSDES WITH CONTINUOUS COEFFICIENT Guangyan Jia and Zhiyong Yu In this paper, we will prove that, if the coefficient $g=g(t,y,z)$ of a BSDE is assumed to be continuous and linear growth in $(y,z)$, then the uniqueness of solution and continuous dependence with respect to $g$ and the terminal value $\xi$ are equivalent. http://arxiv.org/abs/0803.3660 --------------------------------------------------------------- 6837. INTEGRATION WITH RESPECT TO FRACTIONAL LOCAL TIMES WITH HURST INDEX $1/20$ as well as exponential moments. In particular, a formula for the abscissa of convergence of the moment generating function is provided. The results are illustrated with a number of examples at the end of the article. http://arxiv.org/abs/0803.3716 --------------------------------------------------------------- 6841. NORMALLY REFLECTED BROWNIAN MOTION AND SPECTRAL PROPERTIES OF THE NEUMANN LAPLACIAN IN UNBOUNDED DOMAINS Ross Pinsky Let $D\subsetneq R^d$ be an unbounded domain and let $B(t)$ be a Brownian motion in $D$ with normal reflection at the boundary, generated by $\frac12\Delta_N$, where $\Delta_N$ is the Neumann Laplacian on $D$. For a bounded subdomain $U$, let $\tau_U=\inf\{t\ge0:B(t)\in \bar U\}$ denote the first hitting time of $\bar U$. It is well-known that the behavior of the random variable $\tau_U$ determines the global behavior of the Brownian motion--that is, its transience, null recurrence or positive recurrence. In fact, the behavior of $\tau_U$ as $U$ varies over all bounded subdomains determines certain spectral properties of the Neumann Laplacian. In this paper we study the behavior of $\tau_U$ in order to treat both of these issues. Most of the work deals with domains of the form $D=\{(x,z)\in R^{l+m}:|z| 0 is the persistence exponent of the diffusion equation with random initial conditions in spatial dimension d. For n \gg 1 even, the probability that they have no real root on the full real axis decays like n^{-2(\theta(2)+\theta(d))}. For Weyl polynomials and Binomial polynomials, this probability decays respectively like \exp{(-2\theta_{\infty}} \sqrt{n}) and \exp{(-\pi \theta_{\infty} \sqrt{n})} where \theta_{\infty} is such that \theta(d) = 2^{-3/2} \theta_{\infty} \sqrt{d} in large dimension d. We also show that the probability that such polynomials have exactly k roots on a given interval [a,b] has a scaling form given by \exp{(-N_{ab} \tilde \phi(k/N_{ab}))} where N_{ab} is the mean number of real roots in [a,b] and \tilde \phi(x) a universal scaling function. We develop a simple Mean Field (MF) theory reproducing qualitatively these scaling behaviors, and improve systematically this MF approach using the method of persistence with partial survival, which in some cases yields exact results. Finally, we show that the probability density function of the largest absolute value of the real roots has a universal algebraic tail with exponent {-2}. These analytical results are confirmed by detailed numerical computations. http://arxiv.org/abs/0803.4396 --------------------------------------------------------------- 6865. QUENCHED LARGE DEVIATIONS FOR RANDOM WALK IN A RANDOM ENVIRONMENT Atilla Yilmaz We take the point of view of a particle performing random walk with bounded jumps on Z^d in a stationary and ergodic random environment. We prove the quenched large deviation principle (LDP) for the pair empirical measure of the environment Markov chain. By the contraction principle, we deduce the quenched LDP for the mean velocity of the particle and obtain a variational formula for the corresponding rate function. We propose an Ansatz for the minimizer of this formula. We verify this Ansatz for nearest-neighbor walks on Z. As a separate result, we give a probabilistic formula for the ergodic invariant density of the environment Markov chain in the case of ballistic random walk with bounded jumps on Z. http://arxiv.org/abs/0804.0262 --------------------------------------------------------------- 6866. ITO'S FORMULA IN UMD BANACH SPACES AND REGULARITY OF SOLUTIONS OF THE ZAKAI EQUATION Z. Brzezniak and J. M. A. M. van Neerven and M. C. Veraar and L. Weis Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Ito formula is proved which is applied to prove the existence of strong solutions for a class of stochastic evolution equations in UMD Banach spaces. The abstract results are applied to prove regularity in space and time of the solutions of the Zakai equation. http://arxiv.org/abs/0804.0302 --------------------------------------------------------------- 6867. EXISTENCE OF AN INFINITE PARTICLE LIMIT OF STOCHASTIC RANKING PROCESS Kumiko Hattori and Tetsuya Hattori We study a stochastic particle system which models the time evolution of the ranking of books by online bookstores (e.g., Amazon). In this system, particles are lined in a queue. Each particle jumps at random jump times to the top of the queue, and otherwise stays in the queue, being pushed toward the tail every time another particle jumps to the top. In an infinite particle limit, the random motion of each particle between its jumps converges to a deterministic trajectory. (This trajectory is actually observed in the ranking data on web sites.) We prove that the (random) empirical distribution of this particle system converges to a deterministic space-time dependent distribution. A core of the proof is the law of large numbers for {\it dependent} random variables. http://arxiv.org/abs/0804.0321 --------------------------------------------------------------- 6868. ASYMPTOTIC PROPERTIES OF THE MAXIMUM LIKELIHOOD ESTIMATOR FOR STOCHASTIC PARABOLIC EQUATIONS WITH ADDITIVE FRACTIONAL BROWNIAN MOTION Igor Cialenco and Sergey Lototsky and Jan Pospisil A parameter estimation problem is considered for a diagonaliazable stochastic evolution equation using a finite number of the Fourier coefficients of the solution. The equation is driven by additive noise that is white in space and fractional in time with the Hurst parameter $H\geq 1/2$. The objective is to study asymptotic properties of the maximum likelihood estimator as the number of the Fourier coefficients increases. A necessary and sufficient condition for consistency and asymptotic normality is presented in terms of the eigenvalues of the operators in the equation. http://arxiv.org/abs/0804.0407 --------------------------------------------------------------- 6869. PROBABILISTIC INTERPRETATION FOR SYSTEMS OF ISAACS EQUATIONS WITH TWO REFLECTING BARRIERS Rainer Buckdahn and Juan Li In this paper we investigate zero-sum two-player stochastic differential games whose cost functionals are given by doubly controlled reflected backward stochastic differential equations (RBSDEs) with two barriers. For admissible controls which can depend on the whole past and so include, in particular, information occurring before the beginning of the game, the games are interpreted as games of the type "admissible strategy" against "admissible control", and the associated lower and upper value functions are studied. A priori random, they are shown to be deterministic, and it is proved that they are the unique viscosity solutions of the associated upper and the lower Bellman-Isaacs equations with two barriers, respectively. For the proofs we make full use of the penalization method for RBSDEs with one barrier and RBSDEs with two barriers. For this end we also prove new estimates for RBSDEs with two barriers, which are sharper than those in [18]. Furthermore, we show that the viscosity solution of the Isaacs equation with two reflecting barriers not only can be approximated by the viscosity solutions of penalized Isaacs equations with one barrier, but also directly by the viscosity solutions of penalized Isaacs equations without barrier. http://arxiv.org/abs/0804.0311 --------------------------------------------------------------- 6870. HYPERDETERMINANTAL POINT PROCESSES Steven N. Evans and Alex Gottlieb As well as arising naturally in the study of non-intersecting random paths, random spanning trees, and eigenvalues of random matrices, determinantal point processes (sometimes also called fermionic point processes) are relatively easy to simulate and provide a quite broad class of models that exhibit repulsion between points. The fundamental ingredient used to construct a determinantal point process is a kernel giving the pairwise interactions between points: the joint distribution of any number of points then has a simple expression in terms of determinants of certain matrices defined from this kernel. In this paper we initiate the study of an analogous class of point processes that are defined in terms of a kernel giving the interaction between $2M$ points for some integer $M$. The role of matrices is now played by $2M$-dimensional "hypercubic" arrays, and the determinant is replaced by a suitable generalization of it to such arrays -- Cayley's first hyperdeterminant. We show that some of the desirable features of determinantal point processes continue to be exhibited by this generalization. http://arxiv.org/abs/0804.0450 --------------------------------------------------------------- 6871. AN INTRODUCTION TO L\'{E}VY PROCESSES WITH APPLICATIONS IN FINANCE Antonis Papapantoleon These lectures notes aim at introducing L\'{e}vy processes in an informal and intuitive way, accessible to non-specialists in the field. In the first part, we focus on the theory of L\'{e}vy processes. We analyze a `toy' example of a L\'{e}vy process, viz. a L\'{e}vy jump-diffusion, which yet offers significant insight into the distributional and path structure of a L\'{e}vy process. Then, we present several important results about L\'{e}vy processes, such as infinite divisibility and the L\'{e}vy-Khintchine formula, the L\'{e}vy-It\^{o} decomposition, the It\^{o} formula for L\'{e}vy processes and Girsanov's transformation. Some (sketches of) proofs are presented, still the majority of proofs is omitted and the reader is referred to textbooks instead. In the second part, we turn our attention to the applications of L\'{e}vy processes in financial modeling and option pricing. We discuss how the price process of an asset can be modeled using L\'{e}vy processes and give a brief account of market incompleteness. Popular models in the literature are presented and revisited from the point of view of L\'{e}vy processes, and we also discuss three methods for pricing financial derivatives. Finally, some indicative evidence from applications to market data is presented. http://arxiv.org/abs/0804.0482 --------------------------------------------------------------- 6872. CONVERGENCE PROPERTIES OF KEMP'S Q-BINOMIAL DISTRIBUTION Stefan Gerhold and Martin Zeiner We consider Kemp's q-analogue of the binomial distribution. Several convergence results involving the classical binomial, the Heine, the discrete normal, and the Poisson distribution are established. Some of them are q-analogues of classical convergence properties. Besides elementary estimates, we apply Mellin transform asymptotics. http://arxiv.org/abs/0804.0534 --------------------------------------------------------------- 6873. DECENTRALIZED SEARCH WITH RANDOM COSTS Oskar Sandberg A decentralized search algorithm is a method of routing on a random graph that uses only limited, local, information about the realization of the graph. In some random graph models it is possible to define such algorithms which produce short paths when routing from any vertex to any other, while for others it is not. We consider random graphs with random costs assigned to the edges. In this situation, we use the methods of stochastic dynamic programming to create a decentralized search method which attempts to minimize the total cost, rather than the number of steps, of each path. We show that it succeeds in doing so among all decentralized search algorithms which monotonically approach the destination. Our algorithm depends on knowing the expected cost of routing from every vertex to any other, but we show that this may be calculated iteratively, and in practice can be easily estimated from the cost of previous routes and compressed into a small routing table. The methods applied here can also be applied directly in other situations, such as efficient searching in graphs with varying vertex degrees. http://arxiv.org/abs/0804.0577 --------------------------------------------------------------- 6874. ON THE ROLE OF CONVEXITY IN FUNCTIONAL AND ISOPERIMETRIC INEQUALITIES Emanuel Milman This is a continuation of our previous work 0712.4092. It is well known that various isoperimetric inequalities imply their functional ``counterparts'', but in general this is not an equivalence. We show that under certain convexity assumptions (e.g. for log-concave probability measures in Euclidean space), the latter implication can in fact be reversed for very general inequalities, generalizing a reverse form of Cheeger's inequality due to Buser and Ledoux. We develop a coherent single framework for passing between isoperimetric inequalities, Orlicz-Sobolev functional inequalities and capacity inequalities, the latter being notions introduced by Maz'ya and extended by Barthe--Cattiaux--Roberto. As an application, we extend the known results due to the latter authors about the stability of the isoperimetric profile under tensorization, when there is no Central-Limit obstruction. As another application, we show that under our convexity assumptions, $q$-log- Sobolev inequalities ($q \in [1,2]$) are equivalent to an appropriate family of isoperimetric inequalities, extending results of Bakry--Ledoux and Bobkov--Zegarlinski. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the $CD(0,\infty)$ curvature-dimension condition of Bakry--\'Emery. http://arxiv.org/abs/0804.0453 --------------------------------------------------------------- 6875. THE STRUCTURE OF UNICELLULAR MAPS, AND A CONNECTION BETWEEN MAPS OF POSITIVE GENUS AND PLANAR LABELLED TREES Guillaume Chapuy A unicellular map is a map which has only one face. We give a bijection between a dominant subset of rooted unicellular maps of fixed genus and a set of rooted plane trees with distinguished vertices. The bijection applies as well to the case of labelled unicellular maps, which are related to all rooted maps by Marcus and Schaeffer's bijection. This gives an immediate derivation of the asymptotic number of unicellular maps of given genus, and a simple bijective proof of a formula of Lehman and Walsh on the number of triangulations with one vertex. From the labelled case, we deduce an expression of the asymptotic number of maps of genus g with n edges involving the ISE random measure, and an explicit characterization of the limiting profile and radius of random bipartite quadrangulations of genus g in terms of the ISE. http://arxiv.org/abs/0804.0546 --------------------------------------------------------------- 6876. A SYSTEM OF GRABBING PARTICLES RELATED TO GALTON-WATSON TREES Jean Bertoin (DMA and PMA) and Vladas Sidoravicius (UCI and CWI) and Maria Eulalia Vares (CBPF) We consider a system of particles with arms that are activated randomly to grab other particles as a toy model for polymerization. We assume that the following two rules are fulfilled: Once a particle has been grabbed then it cannot be grabbed again, and an arm cannot grab a particle that belongs to its own cluster. We are interested in the shape of a typical polymer in the situation when the initial number of monomers is large and the numbers of arms of monomers are given by i.i.d. random variables. Our main result is a limit theorem for the empirical distribution of polymers, where limit is expressed in terms of a Galton-Watson tree. http://arxiv.org/abs/0804.0726 --------------------------------------------------------------- 6877. CRYPTANALYSIS OF THE ALGEBRAIC ERASER AND SHORT EXPRESSIONS OF PERMUTATIONS AS PRODUCTS Arkadius Kalka and Mina Teicher and and Boaz Tsaban On March 2004, Anshel, Anshel, Goldfeld, and Lemieux introduced the _Algebraic Eraser_ scheme for key agreement over an insecure channel. This scheme is based on semidirect products of algebraic structures, and uses a novel hybrid of infinite and finite noncommutative groups. They also introduced the_Colored Burau Key Agreement Protocol (CBKAP)_, a concrete realization of this scheme. We present an efficient method to extract the shared key out of the public information provided by CBKAP, assuming that the keys are chosen with standard distributions. Our methods come from probabilistic group theory, and seem to have not been used before in cryptanalysis. Of independent interest may be a simple heuristic algorithm we propose for finding short expressions of permutations as products of given random permutations. According to heuristic analysis supported by experiments, our algorithm gives expressions of length O(n^2log n) in running time O(n^4log n). http://arxiv.org/abs/0804.0629 --------------------------------------------------------------- 6878. COMPLEMENTS AND SIGNED DIGIT REPRESENTATIONS: ANALYSIS OF A MULTI-EXPONENTIATION-ALGORITHM OF WU, LOU, LAI AND CHANG Clemens Heuberger and Helmut Prodinger Wu, Lou, Lai and Chang proposed a multi-exponentiation algorithm using binary complements and the non-adjacent form. The purpose of this paper is to show that neither the analysis of the algorithm given by its original proposers nor that by other authors are correct. In fact it turns out that the complement operation does not have significant influence on the performance of the algorithm and can therefore be omitted. http://arxiv.org/abs/0804.0733 --------------------------------------------------------------- 6879. MARKOV JUMP PROCESSES APPROXIMATING A NONSYMMETRIC GENERALIZED DIFFUSION Nedzad Limi\'c Consider a nonsymetric generalized diffusion $X(\cdot)$ in ${\bbR}^d$ generated by the differential operator $A(\msx)=\sum_{ij} \partial_ia_{ij}(\msx)\partial_j +\sum_i b_i(\msx)\partial_i$. In this paper the diffusion process is approximated by Markov jump processes $X_n(\cdot)$ in homogeneous and isotropic grids $G_n \subset {\bbR}^d$ which converge in distribution to diffusion. The generators of $X_n(\cdot)$ are constructed explicitly. Due to the homogeneity and isotropy of grids the proposed method for $d\geq3$ can be applied to processes for which the diffusion tensor $\{a_{ij}(\msx)\}_{11}^{dd}$ fulfills an additional condition. The proposed construction offers a simple method for simulation of sample paths of nonsymetric generalized diffusion. Simulations are carried out in terms of jump processes $X_n(\cdot)$. For $d=2$ the construction can be easily implemented into a computer code. http://arxiv.org/abs/0804.0848 --------------------------------------------------------------- 6880. LIMIT THEOREMS FOR SUBCRITICAL BRANCHING PROCESS IN RANDOM ENVIRONMENT DEPENDING ON THE INITIAL NUMBER OF PARTICLES Vincent Bansaye (PMA) Asymptotic behaviors for subcritical Branching Processes in Random Environment (BPRE) starting with several particles depend on whether the BPRE is strongly subcritical (SS), intermediate subcritical (IS) or weakly subcritical (WS) (see \cite{bpree}). Descendances of particles for BPRE are not independent. In the (SS+IS) case, the asymptotic probability of survival is proportional to the initial number of particles. And conditionally on the survival of the population, only one initial particle survives a.s. These two properties do not hold in the (WS) case and different asymptotics are established, which require to prove new results on random walk with negative drift. We provide an interpretation of these results by characterizing the sequence of environments selected when we condition by the survival of particles. This also raises the problem of the dependence of the Yaglom quasistationary distributions on the initial number of particles and the asymptotic behavior of the Q-process associated with a subcritical BPRE. http://arxiv.org/abs/0804.0853 --------------------------------------------------------------- 6881. CLIQUE PERCOLATION Bela Bollobas and Oliver Riordan Derenyi, Palla and Vicsek introduced the following dependent percolation model, in the context of finding communities in networks. Starting with a random graph $G$ generated by some rule, form an auxiliary graph $G'$ whose vertices are the $k$-cliques of $G$, in which two vertices are joined if the corresponding cliques share $k-1$ vertices. They considered in particular the case where $G=G(n,p)$, and found heuristically the threshold for a giant component to appear in $G'$. Here we give a rigorous proof of this result, as well as many extensions. The model turns out to be very interesting due to the essential global dependence present in $G'$. http://arxiv.org/abs/0804.0867 --------------------------------------------------------------- 6882. ON GAUSSIAN BRUNN-MINKOWSKI INEQUALITIES Franck Barthe (IMT) and Nolwen Huet (IMT) In this paper, we are interested in Gaussian versions of the classical Brunn-Minkowski inequality. We prove in a streamlined way a semigroup version of the Ehrard inequality for $m$ Borel or convex sets based on a previous work by Borell. Our method allows us also to have semigroup proofs of the classical Brascamp-Lieb inequality and of the reverse one which follow exactly the same lines. http://arxiv.org/abs/0804.0886 --------------------------------------------------------------- 6883. SPIKED MODELS IN WISHART ENSEMBLE Dong Wang The spiked model is an important special case of the Wishart ensemble, and a natural generalization of the white Wishart ensemble. Mathematically, it can be defined on three kinds of variables: the real, the complex and the quaternion. For practical application, we are interested in the limiting distribution of the largest sample eigenvalue. We first give a new proof of the result of Baik, Ben Arous and P \'{e}ch\'{e} for the complex spiked model, based on the method of multiple orthogonal polynomials by Bleher and Kuijlaars. Then in the same spirit we present a new result of the rank 1 quaternionic spiked model, proven by combinatorial identities involving quaternionic Zonal polynomials (\alpha = 1/2 Jack polynomials) and skew orthogonal polynomials. We find a phase transition phenomenon for the limiting distribution in the rank 1 quaternionic spiked model as the spiked population eigenvalue increases, and recognize the seemingly new limiting distribution on the critical point as the limiting distribution of the largest sample eigenvalue in the real white Wishart ensemble. Finally we give conjectures for higher rank quaternionic spiked model and the real spiked model. http://arxiv.org/abs/0804.0889 --------------------------------------------------------------- 6884. A LOG-TYPE MOMENT RESULT FOR PERPETUITIES AND ITS APPLICATION TO MARTINGALES IN SUPERCRITICAL BRANCHING RANDOM WALKS Gerold Alsmeyer and Alexander Iksanov Infinite sums of i.i.d. random variables discounted by a multiplicative random walk are called perpetuities and have been studied by many authors. The present paper provides a log-type moment result for such random variables under minimal conditions which is then utilized for the study of related moments of a.s. limits of certain martingales associated with the supercritical branching random walk. The connection, first observed by the second author in [Iksanov, A.M. (2004). Elementary fixed points of the BRW smoothing transforms with infinite number of summands. Stoch. Proc. Appl. 114, 27-50.], arises upon consideration of a size-biased version of the branching random walk originally introduced by Lyons in [Lyons, R.(1997). A simple path to Biggins' martingale convergence for branching random walk. In Athreya, K.B., Jagers, P. (eds.). Classical and Modern Branching Processes, IMA Volumes in Mathematics and its Applications, vol. 84, Springer, Berlin, 217-221.]. We also provide a necessary and sufficient condition for uniform integrability of these martingales in the most general situation which particularly means that the classical (LlogL)-condition is not always needed. http://arxiv.org/abs/0804.0961 --------------------------------------------------------------- 6885. LARGE DEVIATIONS PRINCIPLE FOR PERTURBED CONSERVATION LAWS Mauro Mariani We investigate large deviations for a family of conservative stochastic PDEs (conservation laws) in the asymptotic of jointly vanishing noise and viscosity. We obtain a first large deviations principle in a space of Young measures. The associated rate functional vanishes on a wide set, the so-called set of measure-valued solutions to the limiting conservation law. We therefore investigate a second order large deviations principle, thus providing a quantitative characterization of non-entropic solutions to the conservation law. http://arxiv.org/abs/0804.0997 --------------------------------------------------------------- 6886. PRUNING A L\'EVY CONTINUUM RANDOM TREE Romain Abraham (MAPMO) and Jean-Francois Delmas (CERMICS) and Guillaume Voisin (MAPMO) Given a general critical or sub-critical branching mechanism, we define a pruning procedure of the associated L\'evy continuum random tree. This pruning procedure is defined by adding some marks on the tree, using L\'evy snake techniques. We then prove that the resulting sub-tree after pruning is still a L\'evy continuum random tree. This last result is proved using the exploration process that codes the CRT, a special Markov property and martingale problems for exploration processes. We finally give the joint law under the excursion measure of the lengths of the excursions of the initial exploration process and the pruned one. http://arxiv.org/abs/0804.1027 --------------------------------------------------------------- 6887. MULTIVARIATE FELLER CONDITIONS IN TERM STRUCTURE MODELS: WHY DO(N'T) WE CARE? Peter Spreij and Enno Veerman and Peter Vlaar In this paper, the relevance of the Feller conditions in discrete time macro-finance term structure models is investigated. The Feller conditions are usually imposed on a continuous time multivariate square root process to ensure that the roots have nonnegative arguments. For a discrete time approximate model, the Feller conditions do not give this guarantee. Moreover, in a macro-finance context the restrictions imposed might be economically unappealing. At the same time, it has also been observed that even without the Feller conditions imposed, for a practically relevant term structure model, negative arguments rarely occur. Using models estimated on German data, we compare the yields implied by (approximate) analytic exponentially affine expressions to those obtained through Monte Carlo simulations of very high numbers of sample paths. It turns out that the differences are rarely statistically significant, whether the Feller conditions are imposed or not. Moreover, economically the differences are negligible, as they are always below one basis point. http://arxiv.org/abs/0804.1039 --------------------------------------------------------------- 6888. A FRACTIONAL POISSON EQUATION: EXISTENCE, REGULARITY AND APPROXIMATIONS Marta Sanz-Sol\'e and Iv\'an Torrecilla-Tarantino We consider a stochastic boundary value elliptic problem on a bounded domain $D\subset \mathbb{R}^k$, driven by a fractional Brownian field with Hurst parameter $H=(H_1,...,H_k)\in[{1/2},1[^k$. First we define the stochastic convolution derived from the Green kernel and prove some properties. Using monotonicity methods, we prove existence and uniqueness of solution, along with regularity of the sample paths. Finally, we propose a sequence of lattice approximations and prove its convergence to the solution of the SPDE at a given rate. http://arxiv.org/abs/0804.1108 --------------------------------------------------------------- 6889. STOCHASTIC EVOLUTION EQUATIONS IN UMD BANACH SPACES J. M. A. M. van Neerven and M. C. Veraar and L. Weis We discuss existence, uniqueness, and space-time H\"older regularity for solutions of the parabolic stochastic evolution equation dU(t) = (AU(t) + F(t,U(t))) dt + B(t,U(t)) dW_H(t), t\in [0,\Tend], U(0) = u_0, where $A$ generates an analytic $C_0$-semigroup on a UMD Banach space $E$ and $W_H$ is a cylindrical Brownian motion with values in a Hilbert space $H$. We prove that if the mappings $F:[0,T]\times E\to E$ and $B:[0,T]\times E\to \mathscr{L}(H,E)$ satisfy suitable Lipschitz conditions and $u_0$ is $\F_0$-measurable and bounded, then this problem has a unique mild solution, which has trajectories in $C^\l([0,T];\D((-A)^\theta)$ provided $ \lambda\ge 0$ and $\theta\ge 0$ satisfy $\l+\theta<\frac12$. Various extensions of this result are given and the results are applied to parabolic stochastic partial differential equations. http://arxiv.org/abs/0804.0932 --------------------------------------------------------------- 6890. ANALYSIS OF DISCRETE AND HYBRID STOCHASTIC SYSTEMS BY NONLINEAR CONTRACTION THEORY Quang-Cuong Pham We investigate the stability properties of discrete and hybrid stochastic nonlinear dynamical systems. More precisely, we extend the stochastic contraction theorems (which were formulated for continuous systems) to the case of discrete and hybrid resetting systems. In particular, we show that the mean square distance between any two trajectories of a discrete (or hybrid resetting) contracting stochastic system is upper-bounded by a constant after exponential transients. Using these results, we study the synchronization of noisy nonlinear oscillators coupled by discrete noisy interactions. http://arxiv.org/abs/0804.0934 --------------------------------------------------------------- 6891. A NEW ESTIMATOR FOR THE NUMBER OF SPECIES IN A POPULATION L. Cecconi and A. Gandolfi and C. C. A. Sastri We consider the classic problem of estimating T, the total number of species in a population, from repeated counts in a simple random sample. We look first at the Chao-Lee estimator: we initially show that such estimator can be obtained by reconciling two estimators of the unobserved probability, and then develop a sequence of improvements culminating in a Dirichlet prior Bayesian reinterpretation of the estimation problem. By means of this, we obtain simultaneous estimates of T, of the normalized interspecies variance $ \gamma^2$ and of the parameter $\lambda$ of the prior. Several simulations show that our estimation method is more flexible than several known methods we used as comparison; the only limitation, apparently shared by all other methods, seems to be that it cannot deal with the rare cases in which $\gamma^2 >1$ http://arxiv.org/abs/0804.1030 --------------------------------------------------------------- 6892. THE COMPOUND POISSON DISTRIBUTION AND RETURN TIMES IN DYNAMICAL SYSTEMS N. Haydn and S. Vaienti Previously it has been shown that some classes of mixing dynamical systems have limiting return times distributions that are almost everywhere Poissonian. Here we study the behaviour of return times at periodic points and show that the limiting distribution is a compound Poissonian distribution. We also derive error terms for the convergence to the limiting distribution. We also prove a very general theorem that can be used to establish compound Poisson distributions in many other settings. http://arxiv.org/abs/0804.1032 --------------------------------------------------------------- 6893. BRANCHING PROCESSES IN RANDOM ENVIRONMENT DIE SLOWLY V.Vatutin and A.E.Kyprianou Let $Z_{n,}n=0,1,...,$ be a branching process evolving in the random environment generated by a sequence of iid generating functions $% f_{0}(s),f_{1}(s),...,$ and let $S_{0}=0,S_{k}=X_{1}+...+X_{k},k\geq 1,$ be the associated random walk with $X_{i}=\log f_{i-1}^{\prime}(1),$ $\tau (m,n)$ be the left-most point of minimum of $\left\{S_{k},k\geq 0\right\} $ on the interval $[m,n],$ and $T=\min \left\{k:Z_{k}=0\right\} $. Assuming that the associated random walk satisfies the Doney condition $P(S_{n}>0) \to \rho \in (0,1),n\to \infty ,$ we prove (under the quenched approach) conditional limit theorems, as $n\to \infty $, for the distribution of $Z_{nt},$ $Z_{\tau (0,nt)},$ and $Z_{\tau (nt,n)},$ $t\in (0,1),$ given $T=n$. It is shown that the form of the limit distributions essentially depends on the location of $\tau (0,n)$ with respect to the point $nt.$ http://arxiv.org/abs/0804.1155 --------------------------------------------------------------- 6894. CHAINING TECHNIQUES AND THEIR APPLICATION TO STOCHASTIC FLOWS Michael Scheutzow We review several competing chaining methods to estimate the supremum, the diameter of the range or the modulus of continuity of a stochastic process in terms of tail bounds of their two-dimensional distributions. Then we show how they can be applied to obtain upper bounds for the growth of bounded sets under the action of a stochastic flow. http://arxiv.org/abs/0804.1263 --------------------------------------------------------------- 6895. THE EXPANSION FOR THE OVERLAP FUNCTION Sergio De Carvalho Bezerra In this work, it is proved the complete expansion for the second moment of the overlap function for the Sherrington-Kirkpatrick model. It is a technical result which takes advantage of the cavity method and other induction arguments. http://arxiv.org/abs/0804.1339 --------------------------------------------------------------- 6896. WEAK APPROXIMATION OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS: THE NON LINEAR CASE Arnaud Debussche (IRMAR) We study the error of the Euler scheme applied to a stochastic partial differential equation. We prove that as it is often the case, the weak order of convergence is twice the strong order. A key ingredient in our proof is Malliavin calculus which enables us to get rid of the irregular terms of the error. We apply our method to the case a semilinear stochastic heat equation driven by a space-time white noise. http://arxiv.org/abs/0804.1304 --------------------------------------------------------------- 6897. A FREDHOLM DETERMINANT REPRESENTATION IN ASEP Craig A. Tracy and Harold Widom In previous work the authors found integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice. The dynamics are uniquely determined once the initial state is specified. In this note we restrict our attention to the case of step initial condition with particles at the positive integers, and consider the distribution function for the m'th particle from the left. In the previous work an infinite series of multiple integrals was derived for this distribution. In this note we show that the series can be summed to give a single integral whose integrand involves a Fredholm determinant. http://arxiv.org/abs/0804.1379 --------------------------------------------------------------- 6898. ORDER OF CURRENT VARIANCE AND DIFFUSIVITY IN THE RATE ONE TOTALLY ASYMMETRIC ZERO RANGE PROCESS Marton Balazs and Julia Komjathy We prove that the variance of the current across a characteristic is of order t^{2/3} in a stationary constant rate totally asymmetric zero range process, and that the diffusivity has order t^{1/3}. This is a step towards proving universality of this scaling behavior in the class of one-dimensional interacting systems with one conserved quantity and concave hydrodynamic flux. The proof proceeds via couplings to show the corresponding moment bounds for a second class particle. We build on the methods developed by Balazs- Seppalainen for asymmetric simple exclusion. However, some modifications were needed to handle the larger state space. Our results translate into t^{2/3}- order of variance of the tagged particle on the characteristics of totally asymmetric simple exclusion. http://arxiv.org/abs/0804.1397 --------------------------------------------------------------- 6899. A LOWER BOUND FOR THE PRINCIPAL EIGENVALUE OF THE STOKES OPERATOR IN A RANDOM DOMAIN V. V. Yurinsky This article is dedicated to localization of the principal eigenvalue (PE) of the Stokes operator acting on solenoidal vector fields that vanish outside a large random domain modeling the pore space in a cubic block of porous material with disordered micro-structure. Its main result is an asymptotically deterministic lower bound for the PE of the sum of a low compressibility approximation to the Stokes operator and a small scaled random potential term, which is applied to produce a similar bound for the Stokes PE. The arguments are based on the method proposed by F. Merkl and M. V. W\"{u}trich for localization of the PE of the Schr\"{o}dinger operator in a similar setting. Some additional work is needed to circumvent the complications arising from the restriction to divergence-free vector fields of the class of test functions in the variational characterization of the Stokes PE. http://arxiv.org/abs/0804.1415 --------------------------------------------------------------- 6900. AN ASYMPTOTIC RESULT FOR BROWNIAN POLYMERS Thomas Mountford and Pierre Tarr\`es We consider a model of the shape of a growing polymer introduced by Durrett and Rogers (Probab. Theory Related Fields 92 (1992) 337--349). We prove their conjecture about the asymptotic behavior of the underlying continuous process $X_t$ (corresponding to the location of the end of the polymer at time $t$) for a particular type of repelling interaction function without compact support. http://arxiv.org/abs/0804.1431 --------------------------------------------------------------- 6901. QUENCHED LARGE DEVIATIONS FOR MULTIDIMENSIONAL RANDOM WALK IN RANDOM ENVIRONMENT: A VARIATIONAL FORMULA Jeffrey M. Rosenbluth We take the point of view of the particle in a multidimensional nearest neighbor random walk in random environment (RWRE). We prove a quenched large deviation principle and derive a variational formula for the quenched rate function. Most of the previous results in this area rely on the subadditive ergodic theorem. We employ a different technique which is based on a minimax theorem. Large deviation principles for RWRE have been proven for i.i.d. nestling environments subject to a moment condition and for ergodic uniformly elliptic environments. We assume only that the environment is ergodic and the transition probabilities satisfy a moment condition. http://arxiv.org/abs/0804.1444 --------------------------------------------------------------- 6902. STABILIT\'{E} DU COMPORTEMENT DES MARCHES AL\'{E}ATOIRES SUR UN GROUPE LOCALEMENT COMPACT Driss Gretete Dans cet article nous d\'{e}montrons un th\'{e}or\`{e}me de stabilit \'{e} des probabilit\'{e}s de retour sur un groupe localement compact unimodulaire, s\'{e}parable et compactement engendr\'{e}. Nous d\'{e}montrons que le comportement asymptotique de $F^{*(2n)}(e)$ ne d\'{e}pend pas de la densit\'{e} $F$ sous des hypoth\`{e}ses naturelles. A titre d'exemple nous \'{e}tablissons que la probabilit\'{e} de retour sur une large classe de groupes r \'{e}solubles se comporte comme $\exp(-n^{1/3})$. http://arxiv.org/abs/0804.1461 --------------------------------------------------------------- 6903. ON ESTIMATION AND OPTIMIZATION OF PROBABILITY Xinjia Chen In this paper, we develop a general approach for probabilistic estimation and optimization. An explicit formula is derived for controlling the reliability of probabilistic estimation based on a mixed criterion of absolute and relative errors. By employing the Chernoff bound and the concept of sampling, the minimization of a probabilistic function is transformed into an optimization problem amenable for gradient descendent algorithms. http://arxiv.org/abs/0804.1399 --------------------------------------------------------------- 6904. BOUNDEDNESS OF RIESZ TRANSFORMS FOR ELLIPTIC OPERATORS ON ABSTRACT WIENER SPACES Jan Maas and Jan van Neerven Let (E,H,mu) be an abstract Wiener space and let D_V := VD, where D denotes the Malliavin derivative in the direction of H and V is a closed and densely defined operator from H into another Hilbert space G. Given a bounded operator B on G, coercive on the closure of the range of V, we consider the realisation of the operator D_V* B D_V in L^p(E,mu) for 1}=(0,\infty)$ generated by the $T_j$ which is either $\{1\}$, $\mathbb {R}^{>}$ itself or $r^{\mathbb {Z}}=\{r^n\dvt n\in \mathbb {Z}\}$ for some $r>1$. The first case being trivial, the nontrivial fixed points in the second case are either Weibull distributions or their reciprocal reflections to the negative half line (when represented by random variables), while in the third case further periodic solutions arise. Our analysis builds on the observation that the logarithmic survival function of any fixed point is harmonic with respect to $\varLambda =\sum_{j\geq 1}\delta_{T_j}$, i.e. $\varGamma =\varGamma \star \varLambda$, where $\star$ means multiplicative convolution. This will enable us to apply the powerful Choquet--Deny theorem. http://arxiv.org/abs/0804.1884 --------------------------------------------------------------- 6917. MULTIVARIATE NORMAL APPROXIMATION USING STEIN'S METHOD AND MALLIAVIN CALCULUS Ivan Nourdin (PMA) and Giovanni Peccati (LSTA) and Anthony R \'eveillac (LMA-Rochelle) We combine Stein's method with Malliavin calculus in order to obtain explicit bounds in the multidimensional normal approximation (in the Wasserstein distance) of functionals of Gaussian fields. Our results generalize and refine the main findings by Peccati and Tudor (2005), Nualart and Ortiz-Latorre (2007), Peccati (2007) and Nourdin and Peccati (2007b, 2008); in particular, they apply to approximations by means of Gaussian vectors with an arbitrary, positive definite covariance matrix. Among several examples, we provide an application to a functional version of the Breuer-Major CLT for fields subordinated to a fractional Brownian motion. http://arxiv.org/abs/0804.1889 --------------------------------------------------------------- 6918. ON PROBABILISTIC PARAMETRIC INFERENCE Tomaz Podobnik and Tomi Zivko An objective operational theory of probabilistic parametric inference is formulated without invoking the so-called non-informative prior probability distributions. http://arxiv.org/abs/0804.1905 --------------------------------------------------------------- 6919. MEASURE AND INTEGRAL WITH PURELY ORDINAL SCALES Dieter Denneberg and Michel Grabisch (LIP6) We develop a purely ordinal model for aggregation functionals for lattice valued functions, comprising as special cases quantiles, the Ky Fan metric and the Sugeno integral. For modeling findings of psychological experiments like the reflection effect in decision behaviour under risk or uncertainty, we introduce reflection lattices. These are complete linear lattices endowed with an order reversing bijection like the reflection at 0 on the real interval $[-1,1]$. Mathematically we investigate the lattice of non-void intervals in a complete linear lattice, then the class of monotone interval-valued functions and their inner product. http://arxiv.org/abs/0804.1758 --------------------------------------------------------------- 6920. THE SYMMETRIC SUGENO INTEGRAL Michel Grabisch (LIP6) We propose an extension of the Sugeno integral for negative numbers, in the spirit of the symmetric extension of Choquet integral, also called \Sipos\ integral. Our framework is purely ordinal, since the Sugeno integral has its interest when the underlying structure is ordinal. We begin by defining negative numbers on a linearly ordered set, and we endow this new structure with a suitable algebra, very close to the ring of real numbers. In a second step, we introduce the M\"obius transform on this new structure. Lastly, we define the symmetric Sugeno integral, and show its similarity with the symmetric Choquet integral. http://arxiv.org/abs/0804.1760 --------------------------------------------------------------- 6921. STATIONARY DISTRIBUTIONS FOR DIFFUSIONS WITH INERT DRIFT Richard F. Bass and Krzysztof Burdzy and Zhen-Qing Chen and Martin Hairer Consider a reflecting diffusion in a domain in $R^d$ that acquires drift in proportion to the amount of local time spent on the boundary of the domain. We show that the stationary distribution for the joint law of the position of the reflecting process and the value of the drift vector has a product form. Moreover, the first component is the symmetrizing measure on the domain for the reflecting diffusion without inert drift, and the second component has a Gaussian distribution. We also consider processes where the drift is given in terms of the gradient of a potential. http://arxiv.org/abs/0804.2029 --------------------------------------------------------------- 6922. CRITICAL BEHAVIOR AND THE LIMIT DISTRIBUTION FOR LONG-RANGE ORIENTED PERCOLATION. II: SPATIAL CORRELATION Lung-Chi Chen and Akira Sakai We prove that the Fourier transform of the properly-scaled normalized two-point function for sufficiently spread-out long-range oriented percolation with index \alpha>0 converges to e^{-C|k|^{\alpha\wedge2}} for some positive finite constant C above the upper-critical dimension 2min{\alpha,2}. This answers the open question remained in the previous paper (Chen and Sakai 2008). Moreover, we show that the constant C exhibits a crossover phenomenon at \alpha=2. The proof is based on a new method of estimating fractional moments for the spatial variable of the lace-expansion coefficients. http://arxiv.org/abs/0804.2039 --------------------------------------------------------------- 6923. STOCHASTIC CHAINS WITH MEMORY OF VARIABLE LENGTH Antonio Galves and Eva L\"ocherbach Stochastic chains with memory of variable length constitute an interesting family of stochastic chains of infinite order on a finite alphabet. The idea is that for each past, only a finite suffix of the past, called context, is enough to predict the next symbol. These models were first introduced in the information theory literature by Rissanen (1983) as a universal tool to perform data compression. Recently, they have been used to model up scientific data in areas as different as biology, linguistics and music. This paper presents a personal introductory guide to this class of models focusing on the algorithm Context and its rate of convergence. http://arxiv.org/abs/0804.2050 --------------------------------------------------------------- 6924. SURFACE TENSION IN THE DILUTE ISING MODEL. THE WULFF CONSTRUCTION Marc Wouts (MODAL'x) We study the surface tension and the phenomenon of phase coexistence for the Ising model on $\mathbbm{Z}^d$ ($d \geqslant 2$) with ferromagnetic but random couplings. We prove the convergence in probability (with respect to random couplings) of surface tension and analyze its large deviations : upper deviations occur at volume order while lower deviations occur at surface order. We study the asymptotics of surface tension at low temperatures and relate the quenched value $\tau^q$ of surface tension to maximal flows (first passage times if $d = 2$). For a broad class of distributions of the couplings we show that the inequality $\tau^a \leqslant \tau^q$ -- where $\tau^a$ is the surface tension under the averaged Gibbs measure -- is strict at low temperatures. We also describe the phenomenon of phase coexistence in the dilute Ising model and discuss some of the consequences of the media randomness. All of our results hold as well for the dilute Potts and random cluster models. http://arxiv.org/abs/0804.2208 --------------------------------------------------------------- 6925. STONE-WEIERSTRASS TYPE THEOREMS FOR LARGE DEVIATIONS Henri Comman We give a general version of Bryc's theorem valid on any topological space and with any algebra $\mathcal{A}$ of real-valued continuous functions separating the points, or any well-separating class. In absence of exponential tightness, and when the underlying space is locally compact regular and $\mathcal{A}$ constituted by functions vanishing at infinity, we give a sufficient condition on the functional $\Lambda(\cdot)_{\mid \mathcal{A}}$ to get large deviations with not necessarily tight rate function. We obtain the general variational form of any rate function on a completely regular space; when either exponential tightness holds or the space is locally compact Hausdorff, we get it in terms of any algebra as above. Prohorov-type theorems are generalized to any space, and when it is locally compact regular the exponential tightness can be replaced by a (strictly weaker) condition on $\Lambda(\cdot)_{\mid \mathcal{A}}$. http://arxiv.org/abs/0804.2214 --------------------------------------------------------------- 6926. TWO-DIMENSIONAL MARKOVIAN HOLONOMY FIELDS Thierry L\'evy (DMA) We define a notion of Markov process indexed by curves drawn on a compact surface and taking its values in a compact Lie group. We call such a process a two-dimensional Markovian holonomy field. The prototype of this class of processes, and the only one to have been constructed before the present work, is the canonical process under the Yang-Mills measure, first defined by Ambar Sengupta and later by the author . The Yang-Mills measure sits in the class of Markovian holonomy fields very much like the Brownian motion in the class of Levy processes. We prove that every regular Markovian holonomy field determines a Levy process of a certain class on the Lie group in which it takes its values, and construct, for each Levy process in this class, a Markovian holonomy field to which it is associated. When the Lie group is in fact a finite group, we give an alternative construction of this Markovian holonomy field as the monodromy of a random ramified principal bundle. http://arxiv.org/abs/0804.2230 --------------------------------------------------------------- 6927. A CONSTRUCTIVE PROOF OF THE EXISTENCE OF VITERBI PROCESSES J. Lember and A. Koloydenko Since the early days of digital communication, hidden Markov models (HMMs) have now been also routinely used in speech recognition, processing of natural languages, images, and in bioinformatics. In an HMM $(X_i,Y_i)_{i\ge 1}$, observations $X_1,X_2,...$ are assumed to be conditionally independent given an ``explanatory'' Markov process $Y_1,Y_2,...$, which itself is not observed; moreover, the conditional distribution of $X_i$ depends solely on $Y_i$. Central to the theory and applications of HMM is the Viterbi algorithm to find {\em a maximum a posteriori} (MAP) estimate $q_{1:n}=(q_1,q_2,...,q_n) $ of $Y_{1:n}$ given observed data $x_{1:n}$. Maximum {\em a posteriori} paths are also known as Viterbi paths or alignments. Recently, attempts have been made to study the behavior of Viterbi alignments when $n\to \infty$. Thus, it has been shown that in some special cases a well-defined limiting Viterbi alignment exists. While innovative, these attempts have relied on rather strong assumptions and involved proofs which are existential. This work proves the existence of infinite Viterbi alignments in a more constructive manner and for a very general class of HMMs. http://arxiv.org/abs/0804.2138 --------------------------------------------------------------- 6928. LARGE DEVIATIONS FOR QUANTUM MARKOV SEMIGROUPS ON THE 2 X 2 MATRIX ALGEBRA Henri Comman Let $({\mathcal{T}}_{*t})$ be a predual quantum Markov semigroup acting on the full 2 x 2 matrix algebra and having an absorbing pure state. We prove that for any initial state $\omega$, the net of orthogonal measures representing the net of states $({\mathcal{T}}_{*t}(\omega))$ satisfies a large deviation principle in the pure state space, with a rate function given in terms of the generator, and which does not depend on $\omega$. This implies that $({\mathcal{T}}_{*t}(\omega))$ is faithful for all $t$ large enough. Examples arising in weak coupling limit are studied. http://arxiv.org/abs/0804.2093 --------------------------------------------------------------- 6929. THE SECRECY GRAPH AND SOME OF ITS PROPERTIES Martin Haenggi A new random geometric graph model, the so-called secrecy graph, is introduced and studied. The graph represents a wireless network and includes only edges over which secure communication in the presence of eavesdroppers is possible. The underlying point process models considered are lattices and Poisson point processes. In the lattice case, analogies to standard bond and site percolation can be exploited to determine percolation thresholds. In the Poisson case, the node degrees are determined and percolation is studied using analytical bounds and simulations. It turns out that a small density of eavesdroppers already has a drastic impact on the connectivity of the secrecy graph. http://arxiv.org/abs/0804.2249 --------------------------------------------------------------- 6930. BOUNDS FOR THE LOSS PROBABILITIES OF LARGE LOSS QUEUEING SYSTEMS Vyacheslav M. Abramov The aim of this paper is to establish the bounds for the least root of the functional equation $x=\hat{G}(\mu-\mu x)$, where $\hat{G}(s)$ is the Laplace-Stieltjes transform of an unknown probability distribution function $G(x)$ of a positive random variable having the first two moments $ \frak{g}_1$ and $\frak{g}_2$, and $\mu$ is a positive parameter satisfying the condition $\mu\frak{g}_1>1$. The additional information characterizing $G(x)$ is an empirical probability distribution function ${G}_{\mathrm{emp}}(x)$, and it is assumed that the distance in the uniform (Kolmogorov) metric between $G(x)$ and ${G}_{\mathrm{emp}}(x)$ is not greater than $\kappa$. The obtained bounds for the positive least root of the functional equation $x=\hat{G}(\mu-\mu x)$ are then used to find the asymptotic bounds for the loss probabilities in certain queueing systems with a large number of waiting places, when only an empirical probability distribution function of an interarrival or service time is known. http://arxiv.org/abs/0804.2310 --------------------------------------------------------------- 6931. SPECIAL INVITED PAPER. LARGE DEVIATIONS S. R. S. Varadhan This paper is based on Wald Lectures given at the annual meeting of the IMS in Minneapolis during August 2005. It is a survey of the theory of large deviations. http://arxiv.org/abs/0804.2330 --------------------------------------------------------------- 6932. ALMOST SURE EULER HYDRODYNAMICS OF ONE-DIMENSIONAL ATTRACTIVE PARTICLE SYSTEMS C. Bahadoran (1) and H. Guiol (2) and K. Ravishankar (3) and E. Saada (4) ((1) Univ. Clermont-Ferrand France, (2) Grenoble Univ. France, (3) SUNY USA, (4) CNRS-Rouen France) We consider attractive irreducible conservative particle systems on Z, with at most K particles per site, for which no explicit invariant measures are required. We suppose that jumps have a finite positive first moment. In Bahadoran et al. (2006) we proved, under finite range hypothesis, that for such systems the hydrodynamic limit under Euler scaling exists, and is given by the entropy solution of a scalar conservation law with Lipschitz- continuous flux. Here, by a refinement of our method, we obtain an almost sure hydrodynamic limit, when starting from: i) any shock profile (Riemann hydrodynamics); ii) any general initial profile, but with finite range assumption. http://arxiv.org/abs/0804.2345 --------------------------------------------------------------- 6933. ALMOST-SURE GROWTH RATE OF GENERALIZED RANDOM FIBONACCI SEQUENCES Elise Janvresse (LMRS) and Beno\^it Rittaud (IG and LMPT) and Thierry De La Rue (LMRS) We study the generalized random Fibonacci sequences defined by their first nonnegative terms and for $n\ge 1$, $F_{n+2} = \lambda F_{n+1} \pm F_{n}$ (linear case) and $\widetilde F_{n+2} = |\lambda \widetilde F_{n+1} \pm \widetilde F_{n}|$ (non-linear case), where each $\pm$ sign is independent and either $+$ with probability $p$ or $-$ with probability $1-p$ ($0=2 and 0 < p <= 1, we prove that the expected value of g_n grows exponentially fast. When \lambda = \lambda_k = 2 cos(\pi/k) for some fixed integer k>2, we show that the expected value of g_n grows exponentially fast for p>(2-\lambda_k)/4 and give an algebraic expression for the growth rate. The involved methods extend (and correct) those introduced in a previous paper by the second author. http://arxiv.org/abs/0804.2400 --------------------------------------------------------------- 6935. DYNAMICAL LARGE DEVIATIONS FOR THE BOUNDARY DRIVEN WEAKLY ASYMMETRIC EXCLUSION PROCESS Lorenzo Bertini and Claudio Landim and Mustapha Mourragui We consider the weakly asymmetric exclusion process on a bounded interval with particles reservoirs at the endpoints. The hydrodynamic limit for the empirical density, obtained in the diffusive scaling, is given by the viscous Burgers equation with Dirichlet boundary conditions. We prove the associated dynamical large deviations principle. http://arxiv.org/abs/0804.2458 --------------------------------------------------------------- 6936. ON THE PERMANENT OF RANDOM BERNOULLI MATRICES T. Tao and V. Vu We show that the permanent of an $n \times n$ matrix with iid Bernoulli entries $\pm 1$ is of magnitude $n^{({1/2}+o(1))n}$ with probability $1-o(1)$. In particular, it is almost surely non-zero. http://arxiv.org/abs/0804.2362 --------------------------------------------------------------- 6937. ERROR BOUNDS ON THE NON-NORMAL APPROXIMATION OF HERMITE POWER VARIATIONS OF FRACTIONAL BROWNIAN MOTION Jean-Christophe Breton (LMA-Rochelle) and Ivan Nourdin (PMA) Let $q\geq 2$ be a positive integer, $B$ be a fractional Brownian motion with Hurst index $H\in(0,1)$, $Z$ be an Hermite random variable of index $q $, and $H_q$ denote the Hermite polynomial having degree $q$. For any $n\geq 1$, set $V_n=\sum_{k=0}^{n-1} H_q(B_{k+1}-B_k)$. The aim of the current paper is to derive, in the case when the Hurst index verifies $H>1-1/(2q)$, an upper bound for the total variation distance between the laws $\mathscr{L}(Z_n)$ and $\mathscr{L}(Z)$, where $Z_n$ stands for the correct renormalization of $V_n$ which converges in distribution towards $Z$. Our results should be compared with those obtained recently by Nourdin and Peccati (2007) in the case when $H<1-1/(2q)$, corresponding to the situation where one has normal approximation. http://arxiv.org/abs/0804.2528 --------------------------------------------------------------- 6938. MAX-PLUS DECOMPOSITION OF SUPERMARTINGALES AND CONVEX ORDER. APPLICATION TO AMERICAN OPTIONS AND PORTFOLIO INSURANCE Nicole El Karoui and Asma Meziou We are concerned with a new type of supermartingale decomposition in the Max-Plus algebra, which essentially consists in expressing any supermartingale of class $(\mathcal{D})$ as a conditional expectation of some running supremum process. As an application, we show how the Max-Plus supermartingale decomposition allows, in particular, to solve the American optimal stopping problem without having to compute the option price. Some illustrative examples based on one-dimensional diffusion processes are then provided. Another interesting application concerns the portfolio insurance. Hence, based on the ``Max-Plus martingale,'' we solve in the paper an optimization problem whose aim is to find the best martingale dominating a given floor process (on every intermediate date), w.r.t. the convex order on terminal values. http://arxiv.org/abs/0804.2561 --------------------------------------------------------------- 6939. THE ALLELIC PARTITION FOR COALESCENT POINT PROCESSES Amaury Lambert (CMAP and Fese) Assume that individuals alive at time $t$ in some population can be ranked in such a way that the coalescence times between consecutive individuals are i.i.d. The ranked sequence of these branches is called a coalescent point process. We have shown in a previous work that splitting trees are important instances of such populations. Here, individuals are given DNA sequences, and for a sample of $n$ DNA sequences belonging to distinct individuals, we consider the number $S_n$ of polymorphic sites (sites at which at least two sequences differ), and the number $A_n$ of distinct haplotypes (sequences differing at one site at least). It is standard to assume that mutations arrive at constant rate (on germ lines), and never hit the same site on the DNA sequence. We study the mutation pattern associated to coalescent point processes under this assumption. Here, $S_n$ and $A_n$ grow linearly as $n$ grows, with explicit rate. However, when the branch lengths have infinite expectation, $S_n$ grows more rapidly, e.g. as $n \ln(n)$ for critical birth--death processes. Then, we study the frequency spectrum of the sample, that is, the numbers of polymorphic sites/haplotypes carried by $k$ individuals in the sample. These numbers are shown to grow also linearly with sample size, and we provide simple explicit formulae for mutation frequencies and haplotype frequencies. For critical birth--death processes, mutation frequencies are given by the harmonic series and haplotype frequencies by Fisher logarithmic series. http://arxiv.org/abs/0804.2572 --------------------------------------------------------------- 6940. ON THE ORTHOGONAL POLYNOMIALS ASSOCIATED WITH A L\'EVY PROCESS Josep Llu\'is Sol\'e and Frederic Utzet Let $X=\{X_t, t\ge0\}$ be a c\`{a}dl\`{a}g L\'{e}vy process, centered, with moments of all orders. There are two families of orthogonal polynomials associated with $X$. On one hand, the Kailath--Segall formula gives the relationship between the iterated integrals and the variations of order $n$ of $X$, and defines a family of polynomials $P_1(x_1), P_2(x_1,x_2),...$ that are orthogonal with respect to the joint law of the variations of $X$. On the other hand, we can construct a sequence of orthogonal polynomials $p^{\sigma}_n(x)$ with respect to the measure $\sigma^2\delta_0(dx)+x^2 \nu(dx)$, where $\sigma^2$ is the variance of the Gaussian part of $X$ and $\nu$ its L \'{e}vy measure. These polynomials are the building blocks of a kind of chaotic representation of the square functionals of the L\'{e}vy process proved by Nualart and Schoutens. The main objective of this work is to study the probabilistic properties and the relationship of the two families of polynomials. In particular, the L\'{e}vy processes such that the associated polynomials $P_n(x_1,...,x_n)$ depend on a fixed number of variables are characterized. Also, we give a sequence of L\'{e}vy processes that converge in the Skorohod topology to $X$, such that all variations and iterated integrals of the sequence converge to the variations and iterated integrals of $X $. http://arxiv.org/abs/0804.2585 --------------------------------------------------------------- 6941. NONSTANDARD LIMIT THEOREM FOR INFINITE VARIANCE FUNCTIONALS Allan Sly and Chris Heyde We consider functionals of long-range dependent Gaussian sequences with infinite variance and obtain nonstandard limit theorems. When the long- range dependence is strong enough, the limit is a Hermite process, while for weaker long-range dependence, the limit is $\alpha$-stable L\'{e}vy motion. For the critical value of the long-range dependence parameter, the limit is a sum of a Hermite process and $\alpha$-stable L\'{e}vy motion. http://arxiv.org/abs/0804.2588 --------------------------------------------------------------- 6942. SHAPE TRANSITION UNDER EXCESS SELF-INTERSECTIONS FOR TRANSIENT RANDOM WALK Amine Asselah We reveal a phenomenon of transition in the geometry of a transient simple random walk forced to realize an excess q-fold self-intersection, as the strength parameter, q, is continuously increased. Also, as an application of our approach, we establish a central limit theorem for the q-fold self-intersection in dimension 4 ore more. http://arxiv.org/abs/0804.2616 --------------------------------------------------------------- 6943. ON THE PROBABILISTIC DESCRIPTION OF A MULTIPARTITE CORRELATION EXPERIMENT WITH ARBITRARY NUMBERS OF SETTINGS AND OUTCOMES PER SITE Elena R. Loubenets We consistently formalize the probabilistic description of multipartite joint measurements performed on systems of any nature. This allows us: (1) to specify in probabilistic terms the difference between nonsignaling, the Einstein- Podolsky-Rosen (EPR) locality and Bell's locality; (2) to introduce the notion of an LHV model for an S_{1}x...xS_{N}-setting N-partite correlation experiment, with outcomes of any spectral type, discrete or continuous, and to prove both general and specific "quantum" statements on an LHV simulation in an arbitrary multipartite case; (3) to classify LHV models for a multipartite quantum state, in particular, to show that any N-partite quantum state, pure or mixed, admits an Sx1x...x1 -setting LHV description; (4) to evaluate a threshold visibility for a noisy bipartite quantum state to admit an S_{1}xS_ {2}-setting LHV description under any generalized quantum measurements of two parties. In a sequel to this paper, we shall introduce a single general representation incorporating in a unique manner all Bell-type inequalities for either joint probabilities or correlation functions that have been introduced or will be introduced in the literature. http://arxiv.org/abs/0804.2398 --------------------------------------------------------------- 6944. THE ERGODIC DECOMPOSITION OF ASYMPTOTICALLY MEAN STATIONARY RANDOM SOURCES Alexander Schoenhuth It is demonstrated how to represent asymptotically mean stationary (AMS) random sources with values in standard spaces as mixtures of ergodic AMS sources. This an extension of the well known decomposition of stationary sources which has facilitated the generalization of prominent source coding theorems to arbitrary, not necessarily ergodic, stationary sources. Asymptotic mean stationarity generalizes the definition of stationarity and covers a much larger variety of real-world examples of random sources of practical interest. It is sketched how to obtain source coding and related theorems for arbitrary, not necessarily ergodic, AMS sources, based on the presented ergodic decomposition. http://arxiv.org/abs/0804.2487 --------------------------------------------------------------- 6945. POISSON PROCESSES FOR SUSBSYSTEMS OF FINITE TYPE IN SYMBOLIC DYNAMICS J.-R. Chazottes and Z. Coelho and P. Collet Let $\Delta\subsetneq\V$ be a proper subset of the vertices $\V$ of the defining graph of an irreducible and aperiodic shift of finite type $(\Sigma_{A}^{+},\S)$. Let $\Sigma_{\Delta}$ be the subshift of allowable paths in the graph of $\Sigma_{A}^{+}$ which only passes through the vertices of $\Delta$. For a random point $x$ chosen with respect to an equilibrium state $\mu$ of a H\"older potential $\phi$ on $\Sigma_{A}^{+}$, let $\tau_{n} $ be the point process defined as the sum of Dirac point masses at the times $k>0$, suitably rescaled, for which the first $n$-symbols of $\S^k x$ belong to $\Delta$. We prove that this point process converges in law to a marked Poisson point process of constant parameter measure. The scale is related to the pressure of the restriction of $\phi$ to $\Sigma_{\Delta}$ and the parameters of the limit law are explicitly computed. http://arxiv.org/abs/0804.2550 --------------------------------------------------------------- 6946. ON THE ASYMPTOTIC MEASURE OF PERIODIC SUBSYSTEMS OF FINITE TYPE IN SYMBOLIC DYNAMICS J.-R. Chazottes and Z. Coelho and P. Collet Let $\Delta\subsetneq\V$ be a proper subset of the vertices $\V$ of the defining graph of an aperiodic shift of finite type $(\Sigma_{A}^{+}, \S)$. Let $\Delta_{n}$ be the union of cylinders in $\Sigma_{A}^{+}$ corresponding to the points $x$ for which the first $n$-symbols of $x$ belong to $\Delta$ and let $\mu$ be an equilibrium state of a H\"older potential $\phi$ on $\Sigma_{A}^{+}$. We know that $\mu(\Delta_{n})$ converges to zero as $n$ diverges. We study the asymptotic behaviour of $\mu(\Delta_{n})$ and compare it with the pressure of the restriction of $\phi$ to $\Sigma_{\Delta}$. The present paper extends some results in \cite{CCC} to the case when $\Sigma_{\Delta}$ is irreducible and periodic. We show an explicit example where the asymptotic behaviour differs from the aperiodic case. http://arxiv.org/abs/0804.2551 --------------------------------------------------------------- 6947. CHARACTERIZATION OF COMPACT SUBSETS OF $\MATHCAL{A}^P$ WITH RESPECT TO WEAK TOPOLOGY Hirbod Assa In this brief article we characterize the relatively compact subsets of $\mathcal{A}^p$ for the topology $\sigma(\mathcal{A}^p,\mathcal{R}^q)$ (see below), by the weak compact subsets of $L^p$ . The spaces $ \mathcal{R}^q$ endowed with the weak topology induced by $\mathcal{A}^p$, was recently employed to create the convex risk theory of random processes. The weak compact sets of $\mathcal{A}^p$ are important to characterize the so-called Lebesgue property of convex risk measures, to give a complete description of the Makcey topology on $\mathcal{R}^q$ and for their use in the optimization theory. http://arxiv.org/abs/0804.2873 --------------------------------------------------------------- 6948. UNIFORM OBSERVABILITY OF HIDDEN MARKOV MODELS AND FILTER STABILITY FOR UNSTABLE SIGNALS Ramon van Handel A hidden Markov model is called observable if distinct initial laws give rise to distinct laws of the observation process. Observability implies stability of the nonlinear filter when the signal process is tight, but this need not be the case when the signal process is unstable. This paper introduces a stronger notion of uniform observability which guarantees stability of the nonlinear filter in the absence of stability assumptions on the signal. By developing certain uniform approximation properties of convolution operators, we subsequently demonstrate that the uniform observability condition is satisfied for various classes of filtering models with white noise type observations. This includes the case of observable linear Gaussian filtering models, so that standard results on stability of the Kalman filter are obtained as a special case. http://arxiv.org/abs/0804.2885 --------------------------------------------------------------- 6949. THE CONTINUOUS BEHAVIOR OF THE NUMERAIRE PORTFOLIO UNDER SMALL CHANGES IN INFORMATION STRUCTURE, PROBABILISTIC VIEWS AND INVESTMENT CONSTRAINTS Constantinos Kardaras The numeraire portfolio in a financial market is the unique positive wealth process that makes all other nonnegative wealth processes supermartingales, when deflated by it. The numeraire portfolio depends on market characteristics, which include: (a) the information flow available to acting agents, given by a filtration; (b) the statistical evolution of the asset prices and, more generally, the states of nature, given by a probability measure; and (c) possible restrictions that acting agents might be facing on available investment strategies, modeled by a constraints set. In a financial market with continuous-path asset prices, the stable behavior of the numeraire portfolio is established when each of the aforementioned market parameters is changed in an infinitesimal way. http://arxiv.org/abs/0804.2912 --------------------------------------------------------------- 6950. GEODESICS IN LARGE PLANAR MAPS AND IN THE BROWNIAN MAP Jean-Francois Le Gall We study geodesics in the random metric space called the Brownian map, which appears as the scaling limit of large planar maps. In particular, we completely describe geodesics starting from the distinguished point called the root, and we characterize the set of all points that are connected to the root by more than one geodesic. We also prove that points of the Brownian map can be connected to the root by at most three distinct geodesics. Our results have applications to the behavior of geodesics in large planar maps. http://arxiv.org/abs/0804.3012 --------------------------------------------------------------- 6951. ABSOLUTE CONTINUITY FOR SOME ONE-DIMENSIONAL PROCESSES Nicolas Fournier and Jacques Printems We introduce an elementary method for proving the absolute continuity of the time marginals of one-dimensional processes. It is based on a comparison between the Fourier transform of such time marginals with those of the one-step Euler approximation of the underlying process. We obtain some absolute continuity results for stochastic differential equations with H\"older continuous coefficients. Furthermore, we allow such coefficients to be random and to depend on the whole path of the solution. We also show how it can be extended to some stochastic partial differential equations, and to some L\'evy-driven stochastic differential equations. In the cases under study, the Malliavin calculus cannot be used, because the solution in generally not Malliavin-differentiable. http://arxiv.org/abs/0804.3037 --------------------------------------------------------------- 6952. SMALL PARTS IN THE BERNOULLI SIEVE Alexander Gnedin and Alex Iksanov and Uwe Roesler Sampling from a random discrete distribution induced by a `stick- breaking' process is considered. Under a moment condition, it is shown that the asymptotics of the sequence of occupancy numbers, and of the small- parts counts (singletons, doubletons, etc) can be read off from a limiting model involving a unit Poisson point process and a self-similar renewal process on the halfline. http://arxiv.org/abs/0804.3052 --------------------------------------------------------------- 6953. REFINED CONVERGENCE FOR THE BOOLEAN MODEL Pierre Calka (MAP5) and Julien Michel (UMPA-ENSL) and Katy Paroux (LM-Besan\c{c}on, IRISA) In a previous work, two of the authors proposed a new proof of a well known convergence result for the scaled elementary connected vacant component in the high intensity Boolean model towards the Crofton cell of the Poisson hyperplane process. In this paper, we consider the particular case of the two- dimensional Boolean model where the grains are discs with random radii. We investigate the second-order term in this convergence when the Boolean model and the Poisson line process are coupled on the same probability space. A precise coupling between the Boolean model and the Poisson line process is first established, a result of directional convergence in distribution for the difference of the two sets involved is derived as well. http://arxiv.org/abs/0804.3088 --------------------------------------------------------------- 6954. A CHARACTERIZATION OF DIMENSION FREE CONCENTRATION IN TERMS OF TRANSPORTATION INEQUALITIES Nathael Gozlan (LAMA) The aim of this paper is to show that a probability measure concentrates independently of the dimension like a gaussian measure if and only if it verifies Talagrand's $\T_2$ transportation-cost inequality. This theorem permits us to give a new and very short proof of a result of Otto and Villani. Generalizations to other types of concentration are also considered. In particular, one shows that the Poincar\'e inequality is equivalent to a certain form of dimension free exponential concentration. The proofs of these results rely on simple Large Deviations techniques. http://arxiv.org/abs/0804.3089 --------------------------------------------------------------- 6955. A SIMPLE SAMPLE SIZE FORMULA FOR ESTIMATING MEANS OF POISSON RANDOM VARIABLES Xinjia Chen In this paper, we derive an explicit sample size formula based a mixed criterion of absolute and relative errors for estimating means of Poisson random variables. http://arxiv.org/abs/0804.3033 --------------------------------------------------------------- 6956. ANCESTRAL PROCESS AND DIFFUSION MODEL WITH SELECTION Shuhei Mano The ancestral selection graph in population genetics introduced by Krone and Neuhauser (1997) is an analogue to the coalescent genealogy. The number of ancestral particles, backward in time, of a sample of genes is an ancestral process, which is a birth and death process with quadratic death and linear birth rate. In this paper an explicit form of the number of ancestral particle is obtained, by using the density of the allele frequency in the corresponding diffusion model obtained by Kimura (1955). It is shown that fixation is convergence of the ancestral process to the stationary measure. The time to fixation of an allele is studied in terms of the ancestral process. http://arxiv.org/abs/0804.2696 --------------------------------------------------------------- 6957. BIPOLARIZATION OF POSETS AND NATURAL INTERPOLATION Michel Grabisch (CES) and Christophe Labreuche (TRT) The Choquet integral w.r.t. a capacity can be seen in the finite case as a parsimonious linear interpolator between vertices of $[0,1]^n$. We take this basic fact as a starting point to define the Choquet integral in a very general way, using the geometric realization of lattices and their natural triangulation, as in the work of Koshevoy. A second aim of the paper is to define a general mechanism for the bipolarization of ordered structures. Bisets (or signed sets), as well as bisubmodular functions, bicapacities, bicooperative games, as well as the Choquet integral defined for them can be seen as particular instances of this scheme. Lastly, an application to multicriteria aggregation with multiple reference levels illustrates all the results presented in the paper. http://arxiv.org/abs/0804.2819 --------------------------------------------------------------- 6958. HITTING TIME STATISTICS AND EXTREME VALUE THEORY Ana Cristina Moreira Freitas and Jorge Milhazes Freitas and Mike Todd We consider discrete time dynamical system and show the link between Hitting Time Statistics (the distribution of the first time points land in asymptotically small sets) and Extreme Value Theory (distribution properties of the partial maximum of stochastic processes). This relation allows to study Hitting Time Statistics with tools from Extreme Value Theory, and vice versa. We apply these results to non-uniformly hyperbolic systems and prove that a multimodal map with an absolutely continuous invariant measure must satisfy the classical extreme value laws (with no extra condition on the speed of mixing, for example). We extend these ideas to the subsequent returns to the asymptotically small sets, linking the Poisson statistics of both processes. http://arxiv.org/abs/0804.2887 --------------------------------------------------------------- 6959. SHUFFLING ALGORITHM FOR BOXED PLANE PARTITIONS Alexei Borodin and Vadim Gorin We introduce discrete time Markov chains that preserve uniform measures on boxed plane partitions. Elementary Markov steps change the size of the box from (a x b x c) to ((a-1) x (b+1) x c) or ((a+1) x (b-1) x c). Algorithmic realization of each step involves O((a+b)c) operations. One application is an efficient perfect random sampling algorithm for uniformly distributed boxed plane partitions. Trajectories of our Markov chains can be viewed as random point configurations in the three-dimensional lattice. We compute the bulk limits of the correlation functions of the resulting random point process on suitable two-dimensional sections. The limiting correlation functions define a two-dimensional determinantal point processes with certain Gibbs properties. http://arxiv.org/abs/0804.3071 --------------------------------------------------------------- 6960. INTERLACED PROCESSES ON THE CIRCLE Anthony P. Metcalfe and Neil O'Connell and Jon Warren When two Markov operators commute, it suggests that we can couple two copies of one of the corresponding processes. We explicitly construct a number of couplings of this type for a commuting family of Markov processes on the set of conjugacy classes of the unitary group, using a dynamical rule inspired by the RSK algorithm. Our motivation for doing this is to develop a parallel programme, on the circle, to some recently discovered connections in random matrix theory between reflected and conditioned systems of particles on the line. One of the Markov chains we consider gives rise to a family of Gibbs measures on `bead configurations' on the infinite cylinder. We show that these measures have determinantal structure and compute the corresponding space-time correlation kernel. http://arxiv.org/abs/0804.3142 --------------------------------------------------------------- 6961. CRAM\'{E}R ASYMPTOTICS FOR FINITE TIME FIRST PASSAGE PROBABILITIES OF GENERAL L\'{E}VY PROCESSES Zbigniew Palmowski and Martijn Pistorius We derive the exact asymptotics of $P(\sup_{u\leq t}X(u) > x)$ if $x$ and $t$ tend to infinity with $x/t$ constant, for a L\'{e}vy process $X$ that admits exponential moments. The proof is based on a renewal argument and a two-dimensional renewal theorem of H\"{o}glund (1990). http://arxiv.org/abs/0804.3169 --------------------------------------------------------------- 6962. CONVEX RISK MEASURES: LEBESGUE PROPERTY ON ONE PERIOD AND MULTI PERIOD RISK MEASURES AND APPLICATION IN CAPITAL ALLOCATION PROBLEM Hirbod Assa In this work we study the Lebesgue property for convex risk measures on the space of bounded c\`adl\`ag random processes ($\mathcal{R}^\infty$). Lebesgue property has been defined for one period convex risk measures in \cite{Jo} and earlier had been studied in \cite{De} for coherent risk measures. We introduce and study the Lebesgue property for convex risk measures in the multi period framework. We give presentation of all convex risk measures with Lebesgue property on bounded c\`adl\`ag processes. To do that we need to have a complete description of compact sets of $\mathcal{A}^1$. The main mathematical contribution of this paper is the characterization of the compact sets of $\mathcal{A}^p$ (including $\mathcal{A}^1$). At the final part of this paper, we will solve the Capital Allocation Problem when we work with coherent risk measures. http://arxiv.org/abs/0804.3209 --------------------------------------------------------------- 6963. OPTIMAL STOPPING FOR L\'EVY PROCESSES AND AFFINE FUNCTIONS Diana Dorobantu (LSProba) This paper studies an optimal stopping problem for L\'evy processes. We give a justification of the form of the Snell envelope using standard results of optimal stopping. We also justify the convexity of the value function, and without a priori restriction to a particular class of stopping times, we deduce that the smallest optimal stopping time is necessarily a hitting time. We propose a method which allows to obtain the optimal threshold. Moreover this method allows to avoid long calculations of the integro-differential operatorused in the usual proofs. http://arxiv.org/abs/0804.3277 --------------------------------------------------------------- 6964. THE ANTI-SYMMETRIC GUE MINOR PROCESS Peter J. Forrester and Eric Nordenstam Our study is initiated by a multi-component particle system underlying the tiling of a half hexagon by three species of rhombi. In this particle system species $j$ consists of $\lfloor j/2 \rfloor$ particles which are interlaced with neigbouring species. The joint probability density function (PDF) for this particle system is obtained, and is shown in a suitable scaling limit to coincide with the joint eigenvalue PDF for the process formed by the successive minors of anti-symmetric GUE matrices, which in turn we compute from first principles. The correlations for this process are determinantal and we give an explicit formula for the corresponding correlation kernel in terms of Hermite polynomials. Scaling limits of the latter are computed, giving rise to the Airy kernel, extended Airy kernel and bead kernel at the soft edge and in the bulk, as well as a new kernel at the hard edge. http://arxiv.org/abs/0804.3293 --------------------------------------------------------------- 6965. A PROOF OF THE DALANG-MORTON-WILLINGER THEOREM Dmitry B. Rokhlin We give a new proof of the Dalang-Morton-Willinger theorem, relating the no-arbitrage condition in stochastic securities market models to the existence of an equivalent martingale measure with bounded density for a $d$- dimensional stochastic sequence $(S_n)_{n=0}^N$ of stock prices. Roughly speaking, the proof is reduced to the assertion that under the no-arbitrage condition for N=1 and $S\in L^1$ there exists a strictly positive linear fucntional on $L^1$, which is bounded from above on a special subset of the subspace $K \subset L^1$ of investor's gains. http://arxiv.org/abs/0804.3308 --------------------------------------------------------------- 6966. SCALING LIMITS OF A TAGGED PARTICLE IN THE EXCLUSION PROCESS WITH VARIABLE DIFFUSION COEFFICIENT Milton Jara and Patricia Goncalves We prove a law of large numbers and a central limit theorem for a tagged particle in a symmetric simple exclusion process in the one- dimensional lattice with variable diffusion coefficient. The scaling limits are obtained from a similar result for the current through -1/2 for a zero-range process with bond disorder. For the CLT, we prove convergence to a fractional Brownian motion of Hurst exponent 1/4. http://arxiv.org/abs/0804.3018 --------------------------------------------------------------- 6967. ANISOTROPIC GROWTH OF RANDOM SURFACES IN 2+1 DIMENSIONS Patrik L. Ferrari (1) and Alexei Borodin (2) ((1) WIAS-Berlin and (2) Caltech) We construct a family of stochastic growth models in 2+1 dimensions, that belong to the anisotropic KPZ class. Appropriate projections of these models yield 1+1 dimensional growth models in the KPZ class and random tiling models. We show that correlation functions associated to our models have determinantal structure, and we study large time asymptotics for one of the models. The main asymptotic results are: (1) The growing surface has a limit shape that consists of facets interpolated by a curved piece. (2) The one- point fluctuations of the height function in the curved part are asymptotically normal with variance of order ln(t) for time t>>1. (3) There is a map of the (2+1)-dimensional space-time to the upper half-plane H such that on space-like submanifolds the multi-point fluctuations of the height function are asymptotically equal to those of the pullback of the Gaussian free (massless) field on H. http://arxiv.org/abs/0804.3035 --------------------------------------------------------------- 6968. R-BOUNDEDNESS OF SMOOTH OPERATOR-VALUED FUNCTIONS Mark Veraar and Tuomas Hytonen In this paper we study $R$-boundedness of operator families $\mathcal{T}\subset \calL(X,Y)$, where $X$ and $Y$ are Banach spaces. Under cotype and type assumptions on $X$ and $Y$ we give sufficient conditions for $R$-boundedness. In the first part we show that certain integral operator are $R$-bounded. This will be used to obtain $R$-boundedness in the case that $\mathcal{T}$ is the range of an operator-valued function $T:\R^d\to \calL(X,Y)$ which is in a certain Besov space $B^{d/r}_{r,1}(\R^d; \calL(X,Y))$. The results will be applied to obtain $R$-boundedness of semigroups and evolution families, and to obtain sufficient conditions for existence of solutions for stochastic Cauchy problems. http://arxiv.org/abs/0804.3313 --------------------------------------------------------------- 6969. ARR\^ET OPTIMAL POUR LES PROCESSUS DE MARKOV FORTS ET LES FONCTIONS AFFINES Diana Dorobantu (LSProba) In this Note we study optimal stopping problems for strong Markov processes and affine functions. We give a justification of the Snell envelope form using standard results of optimal stopping. We also justify the convexity of the value function, and without a priori restriction to a particular class of stopping times, we deduce that the smallest optimal stopping time is necessarily a hitting time. http://arxiv.org/abs/0804.3496 --------------------------------------------------------------- 6970. RANDOM WALK IN DETERMINISTICALLY CHANGING ENVIRONMENT Dmitry Dolgopyat and Carlangelo Liverani We consider a random walk with transition probabilities weakly dependent on an environment with a deterministic, but strongly chaotic, evolution. We prove that for almost all initial conditions of the environment the walk satisfies the CLT. http://arxiv.org/abs/0804.3497 --------------------------------------------------------------- 6971. PHASE TRANSITION IN THE 1D RANDOM FIELD ISING MODEL WITH LONG RANGE INTERACTION Marzio Cassandro and Enza Orlandi and Pierre Picco (LATP) We study the one dimensional Ising model with ferromagnetic, long range interaction which decays as |i-j|^{-2+a}, 1/2< a<1, in the presence of an external random filed. we assume that the random field is given by a collection of independent identically distributed random variables, subgaussian with mean zero. We show that for temperature and strength of the randomness (variance) small enough with P=1 with respect to the distribution of the random fields there are at least two distinct extremal Gibbs measures. http://arxiv.org/abs/0804.3672 --------------------------------------------------------------- 6972. ON ESTIMATION OF FINITE POPULATION PROPORTION Xinjia Chen In this paper, we derive an explicit sample size formula for estimating the proportion of a finite population. The sample size obtained from the formula ensures a mixed criterion of absolute and relative errors. http://arxiv.org/abs/0804.3779 --------------------------------------------------------------- 6973. THE M/M/1 QUEUE IS BERNOULLI Michael Keane and Neil O'Connell The classical output theorem for the M/M/1 queue, due to Burke (1956), states that the departure process from a stationary M/M/1 queue, in equilibrium, has the same law as the arrivals process, that is, it is a Poisson process. In this paper we show that the associated measure-preserving transformation is metrically isomorphic to a two-sided Bernoulli shift. We also discuss some extensions of Burke's theorem where it remains an open problem to determine if, or under what conditions, the analogue of this result holds. http://arxiv.org/abs/0804.3935 --------------------------------------------------------------- 6974. QUANTUM GROSS LAPLACIAN AND APPLICATIONS Habib Ouerdiane and Samah Horrigue In this paper, we introduce and study a noncommutative extension of the Gross Laplacian, called quantum Gross Laplacian. Then, applying the quantum Gross Laplacian to the particular case where the operator is the multiplication operator, we find a relation between classical and quantum Gross Laplacian. As application, we give explicit solution of linear quantum white noise differential equation. In particular, we give a explicit solution of the quantum Gross heat equation. http://arxiv.org/abs/0804.3938 --------------------------------------------------------------- 6975. THE NOISY VETO-VOTER MODEL: A RECURSIVE DISTRIBUTIONAL EQUATION ON [0,1] Saul Jacka and Marcus Sheehan We study a particular example of a recursive distributional equation (RDE) on the unit interval. We identify all invariant distributions, the corresponding "basins of attraction" and address the issue of endogeny for the associated tree-indexed problem, making use of an extension of a recent result of Warren. http://arxiv.org/abs/0804.3943 --------------------------------------------------------------- 6976. MAXIMUM PROBABILITY AND RELATIVE ENTROPY MAXIMIZATION. BAYESIAN MAXIMUM PROBABILITY AND EMPIRICAL LIKELIHOOD M. Grendar Works, briefly surveyed here, are concerned with two basic methods: Maximum Probability and Bayesian Maximum Probability; as well as with their asymptotic instances: Relative Entropy Maximization and Maximum Non-parametric Likelihood. Parametric and empirical extensions of the latter methods - Empirical Maximum Maximum Entropy and Empirical Likelihood - are also mentioned. The methods are viewed as tools for solving certain ill-posed inverse problems, called Pi-problem, Phi-problem, respectively. Within the two classes of problems, probabilistic justification and interpretation of the respective methods are discussed. http://arxiv.org/abs/0804.3926 --------------------------------------------------------------- 6977. APPROXIMATE CONTROLLABILITY FOR LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS IN INFINITE DIMENSIONS D. Goreac The objective of the paper is to investigate the approximate controllability property of a linear stochastic control system with values in a separable real Hilbert space. In a first step we prove the existence and uniqueness for the solution of the dual linear backward stochastic differential equation. This equation has the particularity that in addition to an unbounded operator acting on the Y-component of the solution there is still another one acting on the Z-component. With the help of this dual equation we then deduce the duality between approximate controllability and observability. Finally, under the assumption that the unbounded operator acting on the state process of the forward equation is an infinitesimal generator of an exponentially stable semigroup, we show that the generalized Hautus test provides a necessary condition for the approximate controllability. The paper generalizes former results by Buckdahn, Quincampoix and Tessitore (2006) and Goreac (2007) from the finite dimensional to the infinite dimensional case. http://arxiv.org/abs/0804.3893 --------------------------------------------------------------- 6978. INSURANCE, REINSURANCE AND DIVIDEND PAYMENT D. Goreac The aim of this paper is to introduce an insurance model allowing reinsurance and dividend payment. Our model deals with several homogeneous contracts and takes into account the legislation regarding the provisions to be justified by the insurance companies. This translates into some restriction on the (maximal) number of contracts the company is allowed to cover. We deal with a controlled jump process in which one has free choice of retention level and dividend amount. The value function is given as the maximized expected discounted dividends. We prove that this value function is a viscosity solution of some first-order Hamilton-Jacobi-Bellman variational inequality. Moreover, a uniqueness result is provided. http://arxiv.org/abs/0804.3900 --------------------------------------------------------------- 6979. SUSPENSION FLOWS OVER VERSHIK'S AUTOMORPHISMS Alexander I. Bufetov A multiplicative asymptotics is obtained for the deviation of ergodic averages for certain classes of suspension flows over Vershik's automorphisms. http://arxiv.org/abs/0804.3970 --------------------------------------------------------------- 6980. A FAMILY OF SERIES REPRESENTATIONS OF THE MULTIPARAMETER FRACTIONAL BROWNIAN MOTION Anatoliy Malyarenko We derive a family of series representations of the multiparameter fractional Brownian motion in the centred ball of radius $R$ in the $N$- dimensional space $\mathbb{R}^N$. Some known examples of series representations are shown to be the members of the family under consideration. http://arxiv.org/abs/0804.4076 --------------------------------------------------------------- 6981. LOGARITHMIC COMPONENTS OF THE VACANT SET FOR RANDOM WALK ON A DISCRETE TORUS David Windisch This work continues the investigation, initiated in a recent work by Benjamini and Sznitman, of percolative properties of the set of points not visited by a random walk on the discrete torus (Z/NZ)^d up to time uN^d in high dimension d. If u>0 is chosen sufficiently small it has been shown that with overwhelming probability this vacant set contains a unique giant component containing segments of length c_0 log N for some constant c_0 > 0, and this component occupies a non-degenerate fraction of the total volume as N tends to infinity. Within the same setup, we investigate here the complement of the giant component in the vacant set and show that some components consist of segments of logarithmic size. In particular, this shows that the choice of a sufficiently large constant c_0 > 0 is crucial in the definition of the giant component. http://arxiv.org/abs/0804.4097 --------------------------------------------------------------- 6982. GAUSSIAN LIMITS FOR GENERALIZED SPACINGS Yu. Baryshnikov and Mathew D. Penrose and J. E. Yukich Nearest neighbor cells in $R^d$ are used to define coefficients of divergence ($\phi$-divergences) between continuous multivariate samples. For large sample sizes, such distances are shown to be asymptotically normal with a variance depending on the underlying point density. The finite-dimensional distributions of the point measures induced by the coefficients of divergence converge to those of a generalized Gaussian field with a covariance structure determined by the point densities. In $d = 1$, this extends classical central limit theory for sum functions of spacings. The general results yield central limit theorems for logarithmic $k$-spacings, information gain, log-likelihood ratios, and the number of pairs of sample points within a fixed distance of each other. http://arxiv.org/abs/0804.4123 --------------------------------------------------------------- 6983. CONTINUITY PROPERTIES AND INFINITE DIVISIBILITY OF STATIONARY DISTRIBUTIONS OF SOME GENERALISED ORNSTEIN-UHLENBECK PROCESSES Alexander Lindner and Ken-iti Sato Properties of the law $\mu$ of the integral $\int_0^{\infty}c^{- N_{t-}}dY_t$ are studied, where $c>1$ and $\{(N_t,Y_t), t\geq 0\}$ is a bivariate L \'evy process such that $\{N_t \}$ and $\{Y_t \}$ are Poisson processes with parameters $a$ and $b$, respectively. This is the stationary distribution of some generalised Ornstein-Uhlenbeck process. The law $\mu$ is parametrised by $c$, $q$ and $r$, where $p=1-q-r$, $q$, and $r$ are the normalised L \'evy measure of $\{(N_t,Y_t)\}$ at the points $(1,0)$, $(0,1)$ and $(1,1)$, respectively. It is shown that, under the condition that $p>0$ and $q>0$, $\mu_{c,q,r}$ is infinitely divisible if and only if $r\leq pq$. The infinite divisibility of the symmetrisation of $\mu$ is also characterised. The law $\mu$ is either continuous-singular or absolutely continuous, unless $r=1$. It is shown that if $c$ is in the set of Pisot-Vijayaraghavan numbers, which includes all integers bigger than 1, then $\mu$ is continuous-singular under the condition $q>0$. On the other hand, for Lebesgue almost every $c>1$, there are positive constants $C_1$ and $C_2$ such that $\mu$ is absolutely continuous whenever $q\geq C_1 p \geq C_2 r$. For any $c>1$, there is a positive constant $C_3$ such that $\mu$ is continuous-singular whenever $q>0$ and $\max\{q,r\}\leq C_3 p$. Here, if $\{N_t \}$ and $\{Y_t \}$ are independent, then $r=0$ and $q=b/(a+b)$. http://arxiv.org/abs/0804.4258 --------------------------------------------------------------- 6984. LARGE DEVIATIONS FOR RANDOM SPECTRAL MEASURES AND SUM RULES Fabrice Gamboa (IMT) and Alain Rouault (LMA-Versailles) We prove a Large Deviation Principle for the random spec- tral measure associated to the pair $(H_N; e)$ where $H_N$ is sampled in the GUE(N) and e is a fixed unit vector (and more generally in the $\beta$- extension of this model). The rate function consists of two parts. The contribution of the absolutely continuous part of the measure is the reversed Kullback information with respect to the semicircle distribution and the contribution of the singular part is connected to the rate function of the extreme eigenvalue in the GUE. This method is also applied to the Laguerre and Jacobi ensembles, but in thoses cases the expression of the rate function is not so explicit. http://arxiv.org/abs/0804.4322 --------------------------------------------------------------- 6985. ERGODIC OPTIMAL QUADRATIC CONTROL FOR AN AFFINE EQUATION WITH STOCHASTIC AND STATIONARY COEFFICIENTS Giuseppina Guatteri and Federica Masiero We study ergodic quadratic optimal stochastic control problems for an affine state equation with state and control dependent noise and with stochastic coefficients. We assume stationarity of the coefficients and a finite cost condition. We first treat the stationary case and we show that the optimal cost corresponding to this ergodic control problem coincides with the one corresponding to a suitable stationary control problem and we provide a full characterization of the ergodic optimal cost and control. http://arxiv.org/abs/0804.4362 --------------------------------------------------------------- 6986. FRACTIONAL BROWNIAN FLOWS Sreekar Vadlamani We consider stochastic flow on n-dimensional Euclidean space driven by fractional Brownian motion with Hurst parameter H greater than half, and study tangent flow and the growth of the Hausdorff measure of sub-manifolds of the ambient n-dimensional Euclidean space, as they evolve under the flow. The main result is a bound on the rate of (global) growth in terms of the (local) Holder norm of the flow. http://arxiv.org/abs/0804.4376 --------------------------------------------------------------- 6987. EXISTENCE AND REGULARITY OF A NONHOMOGENEOUS TRANSITION MATRIX UNDER MEASURABILITY CONDITIONS Liuer Ye (The School of Mathematics and Computational Science) and Xianping Guo (The School of Mathematics and Computational Science), On \'esimo Hern\'andez-Lerma (Departamento de Matem\'aticas, CINVESTAV-IPN) This paper is about the existence and regularity of the transition probability matrix of a nonhomogeneous continuous-time Markov process with a countable state space. A standard approach to prove the existence of such a transition matrix is to begin with a continuous (in t) and conservative matrix Q(t)=[q_{ij}(t)] of nonhomogeneous transition rates q_{ij}(t), and use it to construct the transition probability matrix. Here we obtain the same result except that the q_{ij}(t) are only required to satisfy a mild measurability condition, and Q(t) may not be conservative. Moreover, the resulting transition matrix is shown to be the minimum transition matrix and, in addition, a necessary and sufficient condition for it to be regular is obtained. These results are crucial in some applications of nonhomogeneous continuous- time Markov processes, such as stochastic optimal control problems and stochastic games, which motivated this work in the first place. http://arxiv.org/abs/0804.4441 --------------------------------------------------------------- 6988. CIRCULAR JACOBI ENSEMBLES AND DEFORMED VERBLUNSKY COEFFICIENTS Paul Bourgade and Ashkan Nikeghbali and Alain Rouault Using spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analogue of the Jacobi ensemble: $$c_{\delta,\beta}^{(n)} \prod_{1\leq k -1/2$. If $e$ is a cyclic vector for a unitary $n\times n$ matrix $U$, the spectral measure of the pair $(U,e)$ is well parameterized by its Verblunsky coefficients $ (\alpha_0,...,\alpha_{n-1})$. We introduce here a deformation $(\gamma_0,...,\gamma_{n-1})$ of these coefficients so that the associated Hessenberg matrix (called GGT) can be decomposed into a product $r(\gamma_0)... r(\gamma_{n-1})$ of elementary reflections parameterized by these coefficients. If $\gamma_0,..., \gamma_{n-1}$ are independent random variables with some remarkable distributions, then the eigenvalues of the GGT matrix follow the circular Jacobi distribution above. These deformed Verblunsky also allow to prove that, in the regime $ \delta = \delta(n)$ with $\delta(n)/n \to \dd$, the spectral measure and the empirical spectral distribution weakly converge to an explicit nontrivial probability measure supported by an arc of the unit circle. http://arxiv.org/abs/0804.4512 --------------------------------------------------------------- 6989. CORDES CONDITIONS AND SOME ALTERNATIVES FOR PARABOLIC EQUATIONS AND DISCONTINUOUS DIFFUSION Nikolai Dokuchaev The paper considers parabolic equations in non-divergent form with discontinuous coefficients at higher derivatives. Their investigation is most complicated because, in general, in the case of discontinuous coefficients, the uniqueness of a solution for nonlinear parabolic or elliptic equations can fail, and there is no a priory estimate for partial derivatives of a solution. In this paper, existence and regularity results are obtained under some Cordes type restrictions on the coefficients. The results are applied to diffusion processes. http://arxiv.org/abs/0804.4519 --------------------------------------------------------------- 6990. OPTIMAL SOLUTION OF INVESTMENT PROBLEMS VIA LINEAR PARABOLIC EQUATIONS GENERATED BY KALMAN FILTER Nikolai Dokuchaev We consider optimal investment problems for a diffusion market model with non-observable random drifts that evolve as an Ito's process. Admissible strategies do not use direct observations of the market parameters, but rather use historical stock prices. For a non-linear problem with a general performance criterion, the optimal portfolio strategy is expressed via the solution of a scalar minimization problem and a linear parabolic equation with coefficients generated by the Kalman filter. http://arxiv.org/abs/0804.4522 --------------------------------------------------------------- 6991. ISING MODELS ON LOCALLY TREE-LIKE GRAPHS Amir Dembo and Andrea Montanari We consider Ising models on graphs that converge locally to trees. Examples include random regular graphs with bounded degree and uniformly random graphs with bounded average degree. We prove that the `cavity' prediction for the limiting free energy per spin is correct for any temperature and external field. Further, local marginals can be approximated by iterating a set of mean field (cavity) equations. Both results are achieved by proving the local convergence of the Boltzmann distribution on the original graph to the Boltzmann distribution on the appropriate infinite random tree. http://arxiv.org/abs/0804.4726 --------------------------------------------------------------- 6992. THE SPECTRUM OF THE RANDOM ENVIRONMENT AND LOCALIZATION OF NOISE Dimitrios Cheliotis and Balint Virag We consider random walk on a mildly random environment on finite transitive d- regular graphs of increasing girth. After scaling and centering, the analytic spectrum of the transition matrix converges in distribution to a Gaussian noise. An interesting phenomenon occurs at d = 2: as the limit graph changes from a regular tree to the integers, the noise becomes localized. http://arxiv.org/abs/0804.4814 --------------------------------------------------------------- 6993. HAMILTONICITY THRESHOLDS IN ACHLIOPTAS PROCESSES Michael Krivelevich and Eyal Lubetzky and Benny Sudakov In this paper we analyze the appearance of a Hamilton cycle in the following random process. The process starts with an empty graph on n labeled vertices. At each round we are presented with K=K(n) edges, chosen uniformly at random from the missing ones, and are asked to add one of them to the current graph. The goal is to create a Hamilton cycle as soon as possible. We show that this problem has three regimes, depending on the value of K. For K=o(\log n), the threshold for Hamiltonicity is (1+o(1))n\log n /(2K), i.e., typically we can construct a Hamilton cycle K times faster that in the usual random graph process. When K=\omega(\log n) we can essentially waste almost no edges, and create a Hamilton cycle in n+o(n) rounds with high probability. Finally, in the intermediate regime where K=\Theta(\log n), the threshold has order n and we obtain upper and lower bounds that differ by a multiplicative factor of 3. http://arxiv.org/abs/0804.4707 From pas at lists.imstat.org Tue Jul 8 02:09:34 2008 From: pas at lists.imstat.org (Probability Abstract Service) Date: Tue, 08 Jul 2008 09:09:34 +0200 Subject: [PAS] Probability Abstracts 104 Message-ID: <8D070538-BEAB-4E0A-9BDF-6793BB551B7B@unimi.it> Probability Abstracts 104 This document contains abstracts 6994-7235 from May-1-2008 to June-30-2008. They have been mailed on July 8th, 2008. This letter can be also found on line at http://pas.imstat.org/Letters/letter_104.shtml --------------------------------------------------------------- 6994. A DISCRETE CONSTRUCTION FOR GAUSSIAN MARKOV PROCESSES Thibaud Taillefumier In the L\'evy construction of Brownian motion, a Haar-derived basis of functions is used to form a finite-dimensional process $W^{N}$ and to define the Wiener process as the almost sure path-wise limit of $W^{N}$ when $N$ tends to infinity. We generalize such a construction to the class of centered Gaussian Markov processes $X$ which can be written $X_{t} = g(t) \cdot \int_{0}^{t} f(t) dW_{t}$ with $f$ and $g$ being continuous functions. We build the finite-dimensional process $X^{N}$ so that it gives an exact representation of the conditional expectation of $X$ with respect to the filtration generated by ${\lbrace X_{k/2^{N}}\rbrace}$ for $0 \leq k \leq 2^{N}$. Moreover, we prove that the process $X^{N}$ converges in distribution toward $X$. http://arxiv.org/abs/0805.0048 --------------------------------------------------------------- 6995. OPTIMAL ROBUST MEAN-VARIANCE HEDGING IN INCOMPLETE FINANCIAL MARKETS N. Lazrieva and T. Toronjadze Optimal B-robust estimate is constructed for multidimensional parameter in drift coefficient of diffusion type process with small noise. Optimal mean-variance robust (optimal V -robust) trading strategy is find to hedge in mean-variance sense the contingent claim in incomplete financial market with arbitrary information structure and misspecified volatility of asset price, which is modelled by multidimensional continuous semimartingale. Obtained results are applied to stochastic volatility model, where the model of latent volatility process contains unknown multidimensional parameter in drift coefficient and small parameter in diffusion term. http://arxiv.org/abs/0805.0122 --------------------------------------------------------------- 6996. COMMUNICATION REQUIREMENTS FOR GENERATING CORRELATED RANDOM VARIABLES Paul Cuff (Stanford University) Two familiar notions of correlation are rediscovered as extreme operating points for simulating a discrete memoryless channel, in which a channel output is generated based only on a description of the channel input. Wyner's "common information" coincides with the minimum description rate needed. However, when common randomness independent of the input is available, the necessary description rate reduces to Shannon's mutual information. This work characterizes the optimal tradeoff between the amount of common randomness used and the required rate of description. http://arxiv.org/abs/0805.0065 --------------------------------------------------------------- 6997. RANDOM WALKS, ARRANGEMENTS, CELL COMPLEXES, GREEDOIDS, AND SELF- ORGANIZING LIBRARIES Anders Bj\"orner The starting point is the known fact that some much-studied random walks on permutations, such as the Tsetlin library, arise from walks on real hyperplane arrangements. This paper explores similar walks on complex hyperplane arrangements. This is achieved by involving certain cell complexes naturally associated with the arrangement. In a particular case this leads to walks on libraries with several shelves. We also show that interval greedoids give rise to random walks belonging to the same general family. Members of this family of Markov chains, based on certain semigroups, have the property that all eigenvalues of the transition matrices are non-negative real and given by a simple combinatorial formula. Background material needed for understanding the walks is reviewed in rather great detail. http://arxiv.org/abs/0805.0083 --------------------------------------------------------------- 6998. RESONANCES FOR A DIFFUSION WITH SMALL NOISE Markus Klein and Pierre-Andr\'e Zitt (MODAL'X) We study resonances for the generator of a diffusion with small noise in $R^d$ :$ L_\epsilon = -\epsilon\Delta + \nabla F \cdot \nabla$, when the potential F grows slowly at infinity (typically as a square root of the norm). The case when F grows fast is well known, and under suitable conditions one can show that there exists a family of exponentially small eigenvalues, related to the wells of F . We show that, for an F with a slow growth, the spectrum is R+, but we can find a family of resonances whose real parts behave as the eigenvalues of the "quick growth" case, and whose imaginary parts are small. http://arxiv.org/abs/0805.0106 --------------------------------------------------------------- 6999. MULTIFRACTAL ANALYSIS IN A MIXED ASYMPTOTIC FRAMEWORK Emmanuel Bacry and Arnaud Gloter and Marc Hoffmann and Jean-Francois Muzy Multifractal analysis of multiplicative random cascades is revisited within the framework of {\em mixed asymptotics}. In this new framework, statistics are estimated over a sample which size increases as the resolution scale (or the sampling period) becomes finer. This allows one to continuously interpolate between the situation where one studies a single cascade sample at arbitrary fine scales and where at fixed scale, the sample length (number of cascades realizations) becomes infinite. We show that scaling exponents of ''mixed'' partitions functions i.e., the estimator of the cumulant generating function of the cascade generator distribution, depends on some ``mixed asymptotic'' exponent $\chi$ respectively above and beyond two critical value $p_ \chi^-$ and $p_\chi^+$. We study the convergence properties of partition functions in mixed asymtotics regime and establish a central limit theorem. These results are shown to remain valid within a general wavelet analysis framework. Their interpretation in terms of Besov frontier are discussed. Moreover, within the mixed asymptotic framework, we establish a ``box-counting'' multifractal formalism that can be seen as a rigorous formulation of Mandelbrot's negative dimension theory. Numerical illustrations of our purpose on specific examples are also provided. http://arxiv.org/abs/0805.0194 --------------------------------------------------------------- 7000. A KHASMINSKII TYPE AVERAGING PRINCIPLE FOR STOCHASTIC REACTION- DIFFUSION EQUATIONS Sandra Cerrai We prove that an averaging principle holds for a general class of stochastic reaction-diffusion systems, having unbounded multiplicative noise, in any space dimension. We show that the classical Khasminskii approach for systems with a finite number of degrees of freedom can be extended to infinite dimensional systems. http://arxiv.org/abs/0805.0294 --------------------------------------------------------------- 7001. AVERAGING PRINCIPLE FOR A CLASS OF STOCHASTIC REACTION- DIFFUSION EQUATIONS Sandra Cerrai and Mark Freidlin We consider the averaging principle for stochastic reaction-diffusion equations. Under some assumptions providing existence of a unique invariant measure of the fast motion with the frozen slow component, we calculate limiting slow motion. The study of solvability of Kolmogorov equations in Hilbert spaces and the analysis of regularity properties of solutions, allow to generalize the classical approach to finite-dimensional problems of this type in the case of SPDE's. http://arxiv.org/abs/0805.0297 --------------------------------------------------------------- 7002. CENTRAL LIMIT THEOREM FOR A CLASS OF LINEAR SYSTEMS Yukio Nagahata and Nobuo Yoshida We consider a class of interacting particle systems with values in $[0,\8)^{\zd}$, of which the binary contact path process is an example. For $d \ge 3$ and under a certain square integrability condition on the total number of the particles, we prove a central limit theorem for the density of the particles, together with upper bounds for the density of the most populated site and the replica overlap. http://arxiv.org/abs/0805.0342 --------------------------------------------------------------- 7003. A LIMIT THEOREM FOR PRODUCTS OF FREE UNITARY OPERATORS Vladislav Kargin This paper establishes necessary and sufficient conditions for the products of freely independent unitary operators to converge in distribution to the uniform law on the unit circle. http://arxiv.org/abs/0805.0374 --------------------------------------------------------------- 7004. THE PLAYER'S EFFECT Ronen Gradwohl and Omer Reingold and Ariel Yadin and Amir Yehudayoff In a function that takes its inputs from various players, the effect of a player measures the variation he can cause in the expectation of that function. In this paper we prove a tight upper bound on the number of players with large effect, a bound that holds even when the players' inputs are only known to be pairwise independent. We also study the effect of a set of players, and show that there always exists a "small" set that, when eliminated, leaves every set with little effect. Finally, we ask whether there always exists a player with positive effect. We answer this question differently in various scenarios, depending on the properties of the function and the distribution of players' inputs. More specifically, we show that if the function is non- monotone or the distribution is only known to be pairwise independent, then it is possible that all players have 0 effect. If the distribution is pairwise independent with minimal support, on the other hand, then there must exist a player with "large" effect. http://arxiv.org/abs/0805.0400 --------------------------------------------------------------- 7005. THE MIXING ADVANTAGE IS LESS THAN 2 Kais Hamza and Peter Jagers and Aidan Sudbury and Daniel Tokarev Corresponding to $n$ independent non-negative random variables $X_1,...,X_n$, are values $M_1,...,M_n$, where each $M_i$ is the expected value of the maximum of $n$ independent copies of $X_i$. We obtain an upper bound to the expected value of the maximum of $X_1,...,X_n$ in terms of $M_1,...,M_n$. This inequality is sharp in the sense that the quantity and its bound can be made as close to each other as we want. We also present related comparison results. http://arxiv.org/abs/0805.0447 --------------------------------------------------------------- 7006. SPECTRAL GAP FOR THE INTERCHANGE PROCESS IN A BOX Ben Morris We show that the spectral gap for the interchange process (and the symmetric exclusion process) in a $d$-dimensional box of side length $L$ is asymptotic to $\pi^2/L^2$. This gives more evidence in favor of Aldous's conjecture that in any graph the spectral gap for the interchange process is the same as the spectral gap for a corresponding continuous-time random walk. Our proof uses a technique that is similar to that used by Handjani and Jungreis, who proved that Aldous's conjecture holds when the graph is a tree. http://arxiv.org/abs/0805.0480 --------------------------------------------------------------- 7007. INTERMITTENCE AND NONLINEAR PARABOLIC STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS Mohammud Foondun and Davar Khoshnevisan We consider nonlinear parabolic SPDEs of the form $\partial_t u=\sL u + \sigma(u)\dot w$, where $\dot w$ denotes space-time white noise, $\sigma:\R\to\R$ is [globally] Lipschitz continuous, and $\sL$ is the $L^2$-generator of a L\'evy process. We present precise criteria for existence as well as uniqueness of solutions. More significantly, we prove that these solutions grow in time with at most a precise exponential rate. We establish also that when $\sigma$ is globally Lipschitz and asymptotically sublinear, the solution to the nonlinear heat equation is ``weakly intermittent,'' provided that the symmetrization of $\sL$ is recurrent and the initial data is sufficiently large. Among other things, our results lead to general formulas for the upper second-moment Liapounov exponent of the parabolic Anderson model for $ \sL$ in dimension $(1+1)$. When $\sL=\kappa\partial_{xx}$ for $\kappa>0$, these formulas agree with the earlier results of statistical physics \cite{Kardar,KrugSpohn,LL63}, and also probability theory \cite{BC,CM94} in the two exactly-solvable cases where $u_0=\delta_0$ and $u_0\equiv 1$. http://arxiv.org/abs/0805.0557 --------------------------------------------------------------- 7008. A MARTINGALE APPROACH TO MINIMAL SURFACES Robert W. Neel We provide a probabilistic approach to studying minimal surfaces in three-dimensional Euclidean space. Following a discussion of the basic relationship between Brownian motion on a surface and minimality of the surface, we introduce a way of coupling Brownian motions on two minimal surfaces. This coupling is then used to study two classes of results in the theory of minimal surfaces, maximum principle-type results, such as weak and strong halfspace theorems and the maximum principle at infinity, and Liouville theorems. http://arxiv.org/abs/0805.0556 --------------------------------------------------------------- 7009. FUNCTIONAL MODERATE DEVIATIONS FOR TRIANGULAR ARRAYS AND APPLICATIONS Florence Merlevede and Magda Peligrad Motivated by the study of dependent random variables by coupling with independent blocks of variables, we obtain first sufficient conditions for the moderate deviation principle in its functional form for triangular arrays of independent random variables. Under some regularity assumptions our conditions are also necessary in the stationary case. The results are then applied to derive moderate deviation principles for linear processes, kernel estimators of a density and some classes of dependent random variables. http://arxiv.org/abs/0805.0617 --------------------------------------------------------------- 7010. RISK AVERSION AND PORTFOLIO SELECTION IN A CONTINUOUS-TIME MODEL Jianming Xia The comparative statics of the optimal portfolios across individuals is carried out for a continuous-time complete market model, where the risky assets price process follows a joint geometric Brownian motion with time- dependent and deterministic coefficients. It turns out that the indirect utility functions inherit the order of risk aversion (in the Arrow-Pratt sense) from the von Neumann-Morgenstern utility functions, and therefore, a more risk- averse agent would invest less wealth (in absolute value) in the risky assets. http://arxiv.org/abs/0805.0618 --------------------------------------------------------------- 7011. PRINCIPAL EIGENVALUE FOR RANDOM WALK AMONG RANDOM TRAPS ON Z^D Jean-Christophe Mourrat Let $(\tau_x)_{x \in \Z^d}$ be i.i.d. random variables with heavy (polynomial) tails. Given $a \in [0,1]$, we consider the Markov process defined by the jump rates $\omega_{x \to y} = {\tau_x}^{-(1-a)} {\tau_y}^a$ between two neighbours $x$ and $y$ in $\Z^d$. We give the asymptotic behaviour of the principal eigenvalue of the generator of this process, with Dirichlet boundary condition. The prominent feature is a phase transition that occurs at some threshold depending on the dimension. Our method relies mainly on results proved in the Appendix, which are of independent interest. They consist of a Gaussian-like upper bound on the transition kernel of any symmetric nearest-neighbour continuous-time random walk on $\Z^d$, provided its jump rates are uniformly bounded from below, together with an upper bound on the Green function when $d \ge 3$. http://arxiv.org/abs/0805.0706 --------------------------------------------------------------- 7012. RANDOM WALK WEAKLY ATTRACTED TO A WALL Jo\"el De Coninck and Fran\c{c}ois Dunlop and Thierry Huillet We consider a random walk $X_n$ in $\Ze_+$, starting at $X_0=x\ge0$, with transition probabilities $$\Pe(X_{n+1}=X_n\pm1|X_n=y\ge1)={1\over2}\mp{\del\over4y+2\del}$$ and $X_{n+1}=1$ whenever $X_n=0$. We prove $\Ee X_n\sim{\rm const.} n^{1-{\del\over2}}$ as $n\nea\infty$ when $\del\in(1,2)$. The proof is based upon the Karlin-McGregor spectral representation, which is made explicit for this random walk. http://arxiv.org/abs/0805.0729 --------------------------------------------------------------- 7013. HYDRODYNAMIC LIMIT FOR A ZERO-RANGE PROCESS IN THE SIERPINSKI GASKET M. Jara We prove that the hydrodynamic limit of a zero-range process evolving in graphs approximating the Sierpinski gasket is given by a nonlinear heat equation. We also prove existence and uniqueness of the hydrodynamic equation by considering a finite-difference scheme. http://arxiv.org/abs/0805.0380 --------------------------------------------------------------- 7014. LEVY PROCESSES AND SCHROEDINGER EQUATION Nicola Cufaro Petroni and Modesto Pusterla We analyze the extension of the well known relation between Brownian motion and Schroedinger equation to the family of Levy processes. We propose a Levy-Schroedinger equation where the usual kinetic energy operator - the Laplacian - is generalized by means of a pseudodifferential operator whose symbol is the logarithmic characteristic of an infinitely divisible law. The Levy-Khintchin formula shows then how to write down this operator in an integro--differential form. When the underlying Levy process is stable we recover as a particular case the recently proposed fractional Schroedinger equation. A few examples are finally given and we find that there are physically relevant models (such as a form of the relativistic Schroedinger equation) that are in the domain of the possible Levy-Schroedinger equations. http://arxiv.org/abs/0805.0503 --------------------------------------------------------------- 7015. LIMIT THEOREMS FOR ADDITIVE C-FREE CONVOLUTION Jiun-Chau Wang In this paper we find necessary and sufficient conditions for the weak convergence of c-free convolution of pairs of measures, where the measures are assumed to be infinitesimal and their support may be unbounded. These results are obtained by complex analytic methods. http://arxiv.org/abs/0805.0607 --------------------------------------------------------------- 7016. THE EFFECT OF CLASSICAL NOISE ON A QUANTUM TWO-LEVEL SYSTEM Jean-Philippe Aguilar (CPT) and Nils Berglund (MAPMO) We consider a quantum two-level system perturbed by classical noise. The noise is implemented as a stationary diffusion process in the off- diagonal matrix elements of the Hamiltonian, representing a transverse magnetic field. We determine the invariant measure of the system and prove its uniqueness. In the case of Ornstein-Uhlenbeck noise, we determine the speed of convergence to the invariant measure. Finally, we determine an approximate one- dimensional diffusion equation for the transition probabilities. The proofs use both spectral-theoretic and probabilistic methods. http://arxiv.org/abs/0805.0869 --------------------------------------------------------------- 7017. CUT POINTS AND DIFFUSIONS IN RANDOM ENVIRONMENT Ivan del Tenno In this article we investigate the asymptotic behavior of a new class of multi-dimensional diffusions in random environment. We introduce cut times in the spirit of the work done by Bolthausen, Sznitman and Zeitouni, see [4], in the discrete setting providing a decoupling effect in the process. This allows us to take advantage of an ergodic structure to derive a strong law of large numbers with possibly vanishing limiting velocity and a central limit theorem under the quenched measure. http://arxiv.org/abs/0805.0886 --------------------------------------------------------------- 7018. BEHAVIOR NEAR THE EXTINCTION TIME IN SELF-SIMILAR FRAGMENTATIONS I: THE STABLE CASE Christina Goldschmidt and B\'en\'edicte Haas (CEREMADE) The stable fragmentation with index of self-similarity $\alpha \in [-1/2,0)$ is derived by looking at the masses of the subtrees formed by discarding the parts of a $(1 + \alpha)^{-1}$--stable continuum random tree below height $t$, for $t \geq 0$. We give a detailed limiting description of the distribution of such a fragmentation, $(F(t), t \geq 0)$, as it approaches its time of extinction, $\zeta$. In particular, we show that $t^{1/\alpha}F((\zeta - t)^+)$ converges in distribution as $t \to 0$ to a non-trivial limit. In order to prove this, we go further and describe the limiting behavior of (a) an excursion of the stable height process (conditioned to have length 1) as it approaches its maximum; (b) the collection of open intervals where the excursion is above a certain level and (c) the ranked sequence of lengths of these intervals. Our principal tool is excursion theory. We also consider the last fragment to disappear and show that, with the same time and space scalings, it has a limiting distribution given in terms of a certain size-biased version of the law of $\zeta$. http://arxiv.org/abs/0805.0967 --------------------------------------------------------------- 7019. ON FINE PROPERTIES OF MIXTURES WITH RESPECT TO CONCENTRATION OF MEASURE AND SOBOLEV TYPE INEQUALITIES Djalil Chafai (IMT and UPTE) and Florent Malrieu (IRMAR) Mixtures are convex combinations of laws. Despite this simple definition, a mixture can be far more subtle than its mixed components. For instance, mixing Gaussian laws may produce a wild potential with multiple wells. We study in the present work fine properties of mixtures with respect to concentration of measure and Gross type functional inequalities. We provide sharp Laplace bounds for Lipschitz functions in the case of generic mixtures, involving a transportation cost diameter of the mixed family. We also provide precise upper bounds for two-components mixtures. Additionally, our analysis of Gross type inequalities for two-components mixtures reveals natural relations with some kind of band isoperimetry and support constrained interpolation via mass transportation. We show that the Poincar\'e constant of a two-components mixture may remain bounded as the mixture proportion goes to 0 or 1 while the Gross constant may surprisingly blow up. Additionally, this counter- intuitive result is not reducible to support disconnections. As far as mixture of distributions are concerned, the Gross inequality is less stable than the sub-Gaussian concentration for Lipschitz functions. We illustrate our results on a gallery of concrete two-components mixtures. http://arxiv.org/abs/0805.0987 --------------------------------------------------------------- 7020. THE EFFECTS OF MASS EXTINCTION EVENTS ON THE GENEALOGY OF A SUBDIVIDED POPULATION Jesse E. Taylor and Amandine Veber We investigate the infinitely many demes limit of the genealogy of a sample of individuals from a subdivided population subject to sporadic mass extinction events. By exploiting a separation of timescales property of Wright's island model, we show that as the number of demes tends to infinity the limiting form of the genealogy can be described in terms of the alternation of instantaneous 'scattering' phases dominated by local demographic processes, and extended 'collecting' phases dominated by global processes. When extinction and recolonization events are local, this genealogy is given by Kingman's coalescent and the scattering phase influences only the overall rate of the process. In contrast, if the vacant demes left by a mass extinction event can be recolonized by individuals emerging from a small number of demes, then the limiting genealogy is a colaescent with simultaneous multiple mergers. In this case, the details of the within-deme population dynamics influence not only the overall rate of the coalescent process, but also the statistics of the complex mergers that can occur within sample genealogies. This study gives some insight into the genealogical consequences of mass extinction in structured populations. http://arxiv.org/abs/0805.1010 --------------------------------------------------------------- 7021. SOME NEW RANDOM FIELD TOOLS FOR SPATIAL ANALYSIS Robert J Adler This is a brief review, in relatively non-technical terms, of recent advances in the theory of random field geometry. These advances have provided a collection of explicit new formulae describing mean values of a variety of geometric characteristics of excursion sets of random fields. As well as a review of the theory, we provide brief descriptions of some of the more interesting applications. http://arxiv.org/abs/0805.1031 --------------------------------------------------------------- 7022. ON THE EIGENSPACES OF LAMPLIGHTER RANDOM WALKS AND PERCOLATION CLUSTERS ON GRAPHS Franz Lehner We show that the Plancherel measure of the lamplighter random walk on a graph coincides with the expected spectral measure of the absorbing random walk on the Bernoulli percolation clusters. In the subcritical regime the spectrum is pure point and we construct a complete set of finitely supported eigenfunctions. http://arxiv.org/abs/0805.0867 --------------------------------------------------------------- 7023. CONCENTRATION OF MEASURE VIA APPROXIMATED BRUNN--MINKOWSKI INEQUALITIES Masayoshi Watanabe We prove that an approximated version of the Brunn--Minkowski inequality with volume distortion coefficient implies a Gaussian concentration-of- measure phenomenon. Our main theorem is applicable to discrete spaces. http://arxiv.org/abs/0805.0902 --------------------------------------------------------------- 7024. EXACTNESS OF MARTINGALE APPROXIMATION AND THE CENTRAL LIMIT THEOREM Dalibor Voln\'y The article is showing sharpness of central limit theorems of Kipnis and Varadhan, Derriennic and Lin, Maxwell and Woodroofe. In the case of the CLT of Derriennic and Lin (for Markov chains with a normal operator) it is shown that the assumption of normality cannot be relaxed. In the case of the CLT of Maxwell and Woodroofe, the example of Peligrad and Utev is improved in the sense of getting a convergence to different laws. http://arxiv.org/abs/0805.1198 --------------------------------------------------------------- 7025. THE FINITE HORIZON OPTIMAL MULTI-MODES SWITCHING PROBLEM: THE VISCOSITY SOLUTION APPROACH Brahim El Asri and Said Hamadene In this paper we show existence and uniqueness of a solution for a system of m variational partial differential inequalities with inter-connected obstacles. This system is the deterministic version of the Verification Theorem of the Markovian optimal m-states switching problem. The switching cost functions are arbitrary. This problem is in relation with the valuation of firms in a financial market. http://arxiv.org/abs/0805.1306 --------------------------------------------------------------- 7026. HYDRODYNAMIC LIMIT OF PARTICLE SYSTEMS WITH LONG JUMPS M. Jara We consider some interacting particle processes with long-range dynamics: the zero-range and exclusion processes with long jumps. We prove that the hydrodynamic limit of these processes corresponds to a (possibly non- linear) fractional heat equation. The scaling in this case is superdiffusive. In addition, we discuss a central limit theorem for a tagged particle on the zero-range process and existence and uniqueness of solutions of the Cauchy problem for the fractional heat equation. http://arxiv.org/abs/0805.1326 --------------------------------------------------------------- 7027. SMALL DEVIATIONS OF GENERAL L\'EVY PROCESSES Frank Aurzada and Steffen Dereich We study the small deviation problem $\log \mathbb{P}(\sup_{t\in[0,1]} |X_t| \leq \epsilon)$, as $\epsilon\to 0$, for general L\'evy processes $X$. The techniques enable us to determine the asymptotic rate for general real- valued L\'evy processes, which we demonstrate with many examples. As a particular consequence, we show that a L\'evy process with non- vanishing Gaussian component has the same (strong) asymptotic small deviation rate as the corresponding Brownian motion. http://arxiv.org/abs/0805.1330 --------------------------------------------------------------- 7028. A NOTE ON THE ENUMERATION OF DIRECTED ANIMALS VIA GAS CONSIDERATIONS Marie Albenque In the literature, most of the results about the enumeration of directed animals on lattices via gas considerations are obtained by a formal passage to the limit of enumeration of directed animals on cyclical versions of the lattice. We provide here a new point of view on this phenomenon. Using the gas construction given introduced by Le Borgne and Marckert, we represent the gas process on the cyclical versions of the lattices as a cyclical Markov chain (roughly speaking, Markov chains conditioned to come back to their starting point). Then we provide a notion of convergence of graphs, such that if $(G_n)$ converges to $G$ then the gas process built on $G_n$ converges in distribution to the gas process on $G$. This gives a general tool to show that gas processes related to animals enumeration are often Markovian on some extracted line of the lattice. We provide examples and computations of new generating functions for directed animals with various sources on some families of lattices. http://arxiv.org/abs/0805.1349 --------------------------------------------------------------- 7029. ADAPTIVE ESTIMATION OF A DISTRIBUTION FUNCTION AND ITS DENSITY IN SUP-NORM LOSS BY WAVELET AND SPLINE PROJECTIONS Evarist Gin\'e and Richard Nickl Given an i.i.d. sample from a distribution $F$ on $\mathbb R$ with uniformly continuous density $p_0$, purely-data driven estimators are constructed that efficiently estimate $F$ in sup-norm loss, and simultaneously estimate $p_0$ at the best possible rate of convergence over H\"{o}lder balls, also in sup-norm loss. The estimators are obtained from applying a model selection procedure close to Lepski's method with random thresholds to projections of the empirical measure onto spaces spanned by wavelets or $B$-splines. Explicit constants in the asymptotic risk of the estimator are obtained, as well as oracle- type inequalities in sup-norm loss. The random thresholds are based on suprema of Rademacher processes indexed by wavelet or spline projection kernels. This requires Bernstein-analogues of the inequalities in Koltchinskii (2006) for the deviation of suprema of empirical processes from their Rademacher symmetrizations. http://arxiv.org/abs/0805.1404 --------------------------------------------------------------- 7030. UNIFORM LIMIT THEOREMS FOR WAVELET DENSITY ESTIMATORS Evarist Gin\'e and Richard Nickl Let $p_n (y)=\sum_k \hat \alpha_{k} \phi(y-k) + \sum_{l=0}^{j_n-1} \sum_k \hat \beta_{lk} 2^{l/2} \psi(2^ly-k)$ be the wavelet density estimator, where $\phi$, $\psi$ are a father and a mother wavelet (with compact support), $\hat \alpha_k$, $\hat \beta_{lk}$ are the empirical wavelet coefficients based on an i.i.d. sample of random variables distributed according to a density $p_0$ on $\mathbb R$, and $j_n \in \mathbb Z$, $j_n \nearrow \infty$. Several uniform limit theorems are proved: First, the almost sure rate of convergence of $\sup_{y \in \mathbb R} |p_n(y)-Ep_n(y)|$ is obtained, and a law of the logarithm for a suitably scaled version of this quantity is established. This implies that $\sup_{y \in \mathbb R} |p_n(y)-p_0(y)|$ attains the optimal almost sure rate of convergence for estimating $p_0$, if $j_n$ is suitably chosen. Second, a uniform central limit theorem as well as strong invariance principles for the distribution function of $p_n$, that is, for the stochastic processes $\sqrt n (F_n^W(s) - F(s))= \sqrt n \int_{-\infty}^s (p_n- p_0), s \in \mathbb R$, are proved; and more generally, uniform central limit theorems for the processes $\sqrt n \int (p_n-p_0)f; f \in \mathcal F$, for other Donsker classes $\mathcal F$ of interest are considered. As a statistical application, it is shown that essentially the same limit theorems can be obtained for the hard thresholding wavelet estimator introduced by Donoho, Johnstone, Kerkyacharian and Picard (1996). http://arxiv.org/abs/0805.1406 --------------------------------------------------------------- 7031. DEGREE-DISTRIBUTION STABILITY OF SCALE-FREE NETWORKS Zhenting Hou and Xiangxing Kong and Dinghua Shi and Guanrong Chen Based on the concept and techniques of first-passage probability in Markov chain theory, this letter provides a rigorous proof for the existence of the steady-state degree distribution of the scale-free network generated by the Barabasi-Albert (BA) model, and mathematically re-derives the exact analytic formulas of the distribution. The approach developed here is quite general, applicable to many other scale-free types of complex networks. http://arxiv.org/abs/0805.1434 --------------------------------------------------------------- 7032. CONDITIONS FOR STOCHASTIC INTEGRABILITY IN UMD BANACH SPACES Jan van Neerven and Mark Veraar and Lutz Weis A detailed theory of stochastic integration in UMD Banach spaces has been developed recently by the authors. The present paper is aimed at giving various sufficient conditions for stochastic integrability. http://arxiv.org/abs/0805.1458 --------------------------------------------------------------- 7033. FLUCTUATIONS OF THE PARTITION FUNCTION IN THE GREM WITH EXTERNAL FIELD Anton Bovier and Anton Klimovsky We study Derrida's generalized random energy model in the presence of uniform external field. We compute the fluctuations of the ground state and of the partition function in the thermodynamic limit for all admissible values of parameters. We find that the fluctuations are described by a hierarchical structure which is obtained by a certain coarse-graining of the initial hierarchical structure of the GREM with external field. We provide an explicit formula for the free energy of the model. We also derive some large deviation results providing an expression for the free energy in a class of models with Gaussian Hamiltonians and external field. Finally, we prove that the coarse-grained parts of the system emerging in the thermodynamic limit tend to have a certain optimal magnetization, as prescribed by strength of external field and by parameters of the GREM. http://arxiv.org/abs/0805.1478 --------------------------------------------------------------- 7034. THE PROBABILITY OF EXCEEDING A PIECEWISE DETERMINISTIC BARRIER BY THE HEAVY-TAILED RENEWAL COMPOUND PROCESS Zbigniew Palmowski and Martijn Pistorius We analyze the asymptotics of crossing a high piecewise linear barriers by a renewal compound process with the subexponential jumps. The study is motivated by ruin probabilities of two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions when the initial reserves of both companies tend to infinity. http://arxiv.org/abs/0805.1631 --------------------------------------------------------------- 7035. HEAT KERNEL ANALYSIS ON INFINITE-DIMENSIONAL HEISENBERG GROUPS Bruce Driver and Maria Gordina We introduce a class of non-commutative Heisenberg like infinite dimensional Lie groups based on an abstract Wiener space. The Ricci curvature tensor for these groups is computed and shown to be bounded. Brownian motion and the corresponding heat kernel measures, $\{\nu_t\}_{t>0},$ are also studied. We show that these heat kernel measures admit: 1) Gaussian like upper bounds, 2) Cameron-Martin type quasi-invariance results, 3) good $L^p$ -- bounds on the corresponding Radon-Nykodim derivatives, 4) integration by parts formulas, and 5) logarithmic Sobolev inequalities. The last three results heavily rely on the boundedness of the Ricci tensor. http://arxiv.org/abs/0805.1650 --------------------------------------------------------------- 7036. QUENCHED AND ANNEALED CRITICAL POINTS IN POLYMER PINNING MODELS Kenneth S. Alexander and Nikos Zygouras We consider a polymer with configuration modeled by the path of a Markov chain, interacting with a potential $u+V_n$ which the chain encounters when it visits a special state 0 at time $n$. The disorder $(V_n)$ is a fixed realization of an i.i.d. sequence. The polymer is pinned, i.e. the chain spends a positive fraction of its time at state 0, when $u$ exceeds a critical value. We assume that for the Markov chain in the absence of the potential, the probability of an excursion from 0 of length $n$ has the form $n^{-c} \phi(n)$ with $c \geq 1$ and $\phi$ slowly varying. Comparing to the corresponding annealed system, in which the $V_n$ are effectively replaced by a constant, it is known that the quenched and annealed critical points differ at all temperatures for $3/22$, but only at low temperatures for $c<3/2$. For high temperatures and $3/23/2$ with arbitrary temperature we provide a new proof that the gap is positive, and extend it to $c=2$. http://arxiv.org/abs/0805.1708 --------------------------------------------------------------- 7037. ON ISOPERIMETRIC INEQUALITIES FOR LOG-CONVEX MEASURES Alexander V. Kolesnikov We study isoperimetric inequalities for measures of the type $ \mu=e^{V} dx$, where $V$ is convex. Using optimal transportation techniques we estimate isoperimetric profiles for a broad class of such measures. We consider many examples and reviel some relations to the hyperbolic geometry and curvature flows. http://arxiv.org/abs/0805.1584 --------------------------------------------------------------- 7038. THE COVARIOGRAM DETERMINES THREE-DIMENSIONAL CONVEX POLYTOPES Gabriele Bianchi The cross covariogram g_{K,L} of two convex sets K, L in R^n is the function which associates to each x in R^n the volume of the intersection of K with L+x. The problem of determining the sets from their covariogram is relevant in stochastic geometry, in probability and it is equivalent to a particular case of the phase retrieval problem in Fourier analysis. It is also relevant for the inverse problem of determining the atomic structure of a quasicrystal from its X-ray diffraction image. The two main results of this paper are that g_{K,K} determines three-dimensional convex polytopes K and that g_{K,L} determines both K and L when K and L are convex polyhedral cones satisfying certain assumptions. These results settle a conjecture of G. Matheron in the class of convex polytopes. Further results regard the known counterexamples in dimension n>=4. We also introduce and study the notion of synisothetic polytopes. This concept is related to the rearrangement of the faces of a convex polytope. http://arxiv.org/abs/0805.1605 --------------------------------------------------------------- 7039. LOG-LEVEL COMPARISON PRINCIPLE FOR SMALL BALL PROBABILITIES A. I. Nazarov We prove a new variant of comparison principle for logarithmic $L_2$- small ball probabilities of Gaussian processes. As an application, we obtain logarithmic small ball asymptotics for some well-known processes with smooth covariances. http://arxiv.org/abs/0805.1773 --------------------------------------------------------------- 7040. CELL CONTAMINATION AND BRANCHING PROCESS IN RANDOM ENVIRONMENT WITH IMMIGRATION Vincent Bansaye (PMA) We consider a branching model for a population of dividing cells infected by parasites. Each cell receives parasites by inheritance from its mother cell and independent contamination from outside the population. Parasites multiply randomly inside the cell and are shared randomly between the two daughter cells when the cell divides. The law of the number of parasites which contaminate a given cell depends only on whether the cell is already infected or not. We determine the asymptotic behavior of the number of parasites in a cell line, which follows a branching process in random environment with state dependent immigration. We then derive a law of large numbers for the asymptotic proportions of cells with a given number of parasites. The main tools are branching processes in random environment and laws of large numbers for Markov tree. http://arxiv.org/abs/0805.1863 --------------------------------------------------------------- 7041. ON THE MARTINGALE PROBEM ASSOCIATED TO THE 2D AND 3D STOCHASTIC NAVIER-STOKES EQUATIONS Giuseppe Da Prato (ENS) and Arnaud Debussche (IRMAR) We consider the martingale problem associated to the Navier-Stokes in dimension 2 or 3. Existence is well known and it has been recently shown that markovian transition semi group associated to these equations can be constructed. We study the Kolmogorov operator associated to these equations. It can be defined formally as a differential operator on an infinite dimensional Hilbert space. It can be also defined in an abstract way as the infinitesimal generator of the transition semi group. We explicit cores for these abstract operators and identify them with the concrete differential operators on these cores. In dimension 2, the core is explicit and we can use a classical argument to prove uniqueness for the martingale problem. In dimension 3, we are only able to exhibit a core which is defined abstractly and does not allow to prove uniqueness for the martingale problem. Instead, we exhibit a core for a modified Kolmogorov operator which enables us to prove uniqueness for the martingale problem up to the time the solutions are regular. http://arxiv.org/abs/0805.1906 --------------------------------------------------------------- 7042. ON ONE PROPERTY OF DISTANCES IN THE INFINITE RANDOM QUADRANGULATION Maxim Krikun (IECN) We show that the Schaeffer's tree for an infinite quadrangulation only changes locally when changing the root of the quadrangulation. This follows from one property of distances in the infinite uniform random quadrangulation. http://arxiv.org/abs/0805.1907 --------------------------------------------------------------- 7043. AGGREGATION OF WEAKLY DEPENDENT DOUBLY STOCHASTIC PROCESSES Lisandro J. Fermin The aim of this paper is to extend the aggregation convergence results given in (Dacunha-Castelle and Fermin 2005, Dacunha-Castelle and Fermin 2008) to doubly stochastic linear and nonlinear processes with weakly dependent innovations. First, we introduce a weak dependence notion for doubly stochastic processes, based in the weak dependence definition given in (Doukhan and Louhichi 1999), and we exhibe several models satisfying this notion, such as: doubly stochastic Volterra processes and doubly stochastic Bernoulli scheme with weakly dependent innovations. Afterwards we derive a central limit theorem for the partial aggregation sequence considering weakly dependent doubly stochastic processes. Finally, show a new SLLN for the covariance function of the partial aggregation process in the case of doubly stochastic Volterra processes with interactive innovations. Keywords: Aggregation, weak dependence, doubly stochastic processes, Volterra processes, Bernoulli shift, TCL, SLLN. http://arxiv.org/abs/0805.1949 --------------------------------------------------------------- 7044. ON A SET OF TRANSFORMATIONS OF GAUSSIAN RANDOM FUNCTIONS A.I. Nazarov We consider a set of one-dimensional transformations of Gaussian random functions. Under natural assumptions we obtain a connection between $L_2$-small ball asymptotics of the transformed function and of the original one. Also the explicit Karhunen -- Lo\'eve expansion is obtained for a proper class of Gaussian processes. http://arxiv.org/abs/0805.1967 --------------------------------------------------------------- 7045. STOCHASTIC CALCULUS FOR CONVOLUTED L\'{E}VY PROCESSES Christian Bender and Tina Marquardt We develop a stochastic calculus for processes which are built by convoluting a pure jump, zero expectation L\'{e}vy process with a Volterra-type kernel. This class of processes contains, for example, fractional L\'{e}vy processes as studied by Marquardt [Bernoulli 12 (2006) 1090--1126.] The integral which we introduce is a Skorokhod integral. Nonetheless, we avoid the technicalities from Malliavin calculus and white noise analysis and give an elementary definition based on expectations under change of measure. As a main result, we derive an It\^{o} formula which separates the different contributions from the memory due to the convolution and from the jumps. http://arxiv.org/abs/0805.2084 --------------------------------------------------------------- 7046. STOCHASTIC ANALYSIS ON GAUSSIAN SPACE APPLIED TO DRIFT ESTIMATION Nicolas Privault and Anthony Reveillac In this paper we consider the nonparametric functional estimation of the drift of Gaussian processes using Paley-Wiener and Karhunen-Lo\`eve expansions. We construct efficient estimators for the drift of such processes, and prove their minimaxity using Bayes estimators. We also construct superefficient estimators of Stein type for such drifts using the Malliavin integration by parts formula and stochastic analysis on Gaussian space, in which superharmonic functionals of the process paths play a particular role. Our results are illustrated by numerical simulations and extend the construction of James-Stein type estimators for Gaussian processes by Berger and Wolper. http://arxiv.org/abs/0805.2002 --------------------------------------------------------------- 7047. THE DBAR STEEPEST DESCENT METHOD FOR ORTHOGONAL POLYNOMIALS ON THE REAL LINE WITH VARYING WEIGHTS K. T.-R. McLaughlin and P. D. Miller We obtain Plancherel-Rotach type asymptotics valid in all regions of the complex plane for orthogonal polynomials with varying weights of the form $e^{-NV(x)}$ on the real line, assuming that $V$ has only two Lipschitz continuous derivatives and that the corresponding equilibrium measure has typical support properties. As an application we extend the universality class for bulk and edge asymptotics of eigenvalue statistics in unitary invariant Hermitian random matrix theory. Our methodology involves developing a new technique of asymptotic analysis for matrix Riemann-Hilbert problems with nonanalytic jump matrices suitable for analyzing such problems even near transition points where the solution changes from oscillatory to exponential behavior. http://arxiv.org/abs/0805.1980 --------------------------------------------------------------- 7048. THE KOLMOGOROV OPERATOR ASSOCIATED TO A BURGERS SPDE IN SPACES OF CONTINUOUS FUNCTIONS Luigi Manca We are concerned with a viscous Burgers equation forced by a perturbation of white noise type. We study the corresponding transition semigroup in a space of continuous functions weighted by a proper potential, and we show that the infinitesimal generator is the closure (with respect to a suitable topology) of the Kolmogorov operator associated to the stochastic equation. In the last part of the paper we use this result to solve the corresponding Fokker-Planck equation. http://arxiv.org/abs/0805.2011 --------------------------------------------------------------- 7049. UNIVERSAL OPTIMAL STOCHASTIC EXPANSIONS Simon J.A. Malham and Anke Wiese We study solutions to nonlinear stochastic differential systems driven by a multi-dimensional Wiener process with non-commuting diffusion vector fields, and no drift. We construct universal optimal solution expansions. They are optimal because the solution series truncated at any order is at least as accurate as the corresponding stochastic Taylor truncation in the mean- square sense. They are universal because this property is independent of the vector fields concerned. This series is the hyperbolic sine of the logarithm of the stochastic Taylor flow. Our proof utilizes the underlying Hopf algebra structure of these series, and a two-alphabet associative algebra of shuffle and concatenation operations that distinguish the coefficients of each term in the series. http://arxiv.org/abs/0805.2340 --------------------------------------------------------------- 7050. PROBABILISTIC REPRESENTATION FOR SOLUTIONS OF AN IRREGULAR POROUS MEDIA TYPE EQUATION Philippe Blanchard and Michael R\"ockner (SFB 705) and Francesco Russo (LAGA) We consider a porous media type equation over all of $\R^d$ with $d = 1$, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. This equation is motivated by some singular behaviour arising in complex self-organized critical systems. One of the main analytic ingredients of the proof, is a new result on uniqueness of distributional solutions of a linear PDE on $\R^1$ with non-continuous coefficients. http://arxiv.org/abs/0805.2383 --------------------------------------------------------------- 7051. N/V-LIMIT FOR LANGEVIN DYNAMICS IN CONTINUUM Florian Conrad and Martin Grothaus We construct an infinite particle/infinite volume Langevin dynamics on the space of configurations in $\R^d$ having velocities as marks. The construction is done via a limiting procedure using $N$-particle dynamics in cubes $(-\lambda,\lambda]^d$ with periodic boundary conditions. A main step to this result is to derive an (improved) Ruelle bound for the canonical correlation functions of $N$-particle systems in $(-\lambda,\lambda]^d$ with periodic boundary conditions. After proving tightness of the laws of finite particle dynamics, the identification of accumulation points as martingale solutions of the Langevin equation is based on a general study of properties of measures on configuration space (and their weak limit) fulfilling a uniform Ruelle bound. Additionally, we prove that the initial/invariant distribution of the constructed dynamics is a tempered grand canonical Gibbs measure. All proofs work for general repulsive interaction potentials $\phi$ of Ruelle type (e.g. the Lennard-Jones potential) and all temperatures, densities and dimensions $d\geq 1$. http://arxiv.org/abs/0805.2518 --------------------------------------------------------------- 7052. OVERCROWDING AND HOLE PROBABILITIES FOR RANDOM ZEROS ON COMPLEX MANIFOLDS Bernard Shiffman and Steve Zelditch and Scott Zrebiec We give asymptotic large deviations estimates for the volume inside a domain U of the zero set of a random holomorphic section of the N-th power of a positive line bundle on a compact Kaehler manifold. In particular, we show that for all $\delta>0$, the probability that this volume differs by more than $\delta N$ from its average value is less than $\exp(-C_{\delta,U}N^{m +1})$, for some constant $C_{\delta,U}>0$. As a consequence, the "hole probability" that a random section does not vanish in U has an upper bound of the form $\exp(-C_{U}N^{m+1})$. http://arxiv.org/abs/0805.2598 --------------------------------------------------------------- 7053. PHASE TRANSITIONS FOR THE GROETH RATE OF LINEAR STOCHASTIC EVOLUTIONS Nobuo Yoshida We consider a simple discrete-time Markov chain with values in $[0,\infty)^{Z^d}$. The Markov chain describes various interesting examples such as oriented percolation, directed polymers in random environment, time discretizations of binary contact path process and the voter model. We study the phase transition for the growth rate of the "total number of particles" in this framework. The main results are roughly as follows: If $d \ge 3$ and the Markov chain is "not too random", then, with positive probability, the growth rate of the total number of particles is of the same order as its expectation. If on the other hand, $d=1,2$, or the Markov chain is "random enough", then the growth rate is slower than its expectation. We also discuss the above phase transition for the dual processes and its connection to the structure of invariant measures for the Markov chain with proper normalization. http://arxiv.org/abs/0805.2652 --------------------------------------------------------------- 7054. PROBABILITY THEORY AND ITS MODELS Paul Humphreys This paper argues for the status of formal probability theory as a mathematical, rather than a scientific, theory. David Freedman and Philip Stark's concept of model based probabilities is examined and is used as a bridge between the formal theory and applications. http://arxiv.org/abs/0805.2801 --------------------------------------------------------------- 7055. DUTCH BOOK IN SIMPLE MULTIVARIATE NORMAL PREDICTION: ANOTHER LOOK Morris L. Eaton In this expository paper we describe a relatively elementary method of establishing the existence of a Dutch book in a simple multivariate normal prediction setting. The method involves deriving a nonstandard predictive distribution that is motivated by invariance. This predictive distribution satisfies an interesting identity which in turn yields an elementary demonstration of the existence of a Dutch book for a variety of possible predictive distributions. http://arxiv.org/abs/0805.2808 --------------------------------------------------------------- 7056. GENERATING UNIFORM RANDOM VECTORS IN $\QTR{BF}{Z}_{P}^{K}$: THE GENERAL CASE Claudio Asci This paper is about the rate of convergence of the Markov chain $X_{n+1}=AX_{n}+B_{n}$ (mod $p$), where $A$ is an integer matrix with nonzero eigenvalues and ${B_{n}}_{n}$ is a sequence of independent and identically distributed integer vectors, with support not parallel to a proper subspace of $Q^{k}$ invariant under $A$. If $|\lambda_{i}|\not=1$ for all eigenvalues $\lambda_{i}$ of $A$, then $n=O((\ln p)^{2}) $ steps are sufficient and $n=O(\ln p)$ steps are necessary to have $X_{n}$ sampling from a nearly uniform distribution. Conversely, if $A$ has the eigenvalues $\lambda_{i}$ that are roots of positive integer numbers, $|\lambda_{1}|=1$ and $|\lambda_{i}| >1$ for all $i\not=1$, then $O(p^{2}) $ steps are necessary and sufficient. http://arxiv.org/abs/0805.2830 --------------------------------------------------------------- 7057. MODERATE DEVIATIONS FOR STATIONARY SEQUENCES OF HILBERT VALUED BOUNDED RANDOM VARIABLES Sophie Dede (PMA) In this paper, we derive the moderate deviation principle for stationary sequences of bounded random variables with values in a Hilbert space. The conditions obtained are expressed in terms of martingale-type conditions. The main tools are martingale approximations and a new Hoeffding inequality for non adpated sequences of Hilbert-valued random variables. Applications to Cramer-Von Mises statistics, functions of linear processes and stable Markov chains are given. http://arxiv.org/abs/0805.2899 --------------------------------------------------------------- 7058. ON ALMOST RANDOMIZING CHANNELS WITH A SHORT KRAUS DECOMPOSITION Guillaume Aubrun (ICJ) For large $d$, we study quantum channels on $\C^d$ obtained by selecting randomly $N$ independent Kraus operators according to a probability measure $\mu$ on the unitary group $\mU(d)$. When $\mu$ is the Haar measure, we show that for $N \succcurlyeq d/\e^2$, such a channel is $\e$-randomizing with high probability, which means that it maps every state within distance $\e/d $ (in operator norm) of the maximally mixed state. This slightly improves on a result by Hayden, Leung, Shor and Winter by optimizing their discretization argument. Moreover, for general $\mu$, we obtain a $\e$-randomizing channel provided $N \succcurlyeq d (\log d)^6/\e^2$. For $d=2^k$ ($k$ qubits), this includes Kraus operators obtained by tensoring $k$ random Pauli matrices. The proof uses recent results on empirical processes in Banach spaces. http://arxiv.org/abs/0805.2900 --------------------------------------------------------------- 7059. EVOLUTION EQUATIONS OF THE PROBABILISTIC GENERALIZATION OF THE VOIGT PROFILE FUNCTION Gianni Pagnini and Francesco Mainardi The spectrum profile that emerges in molecular spectroscopy and atmospheric radiative transfer as the combined effect of Doppler and pressure broadenings is known as the Voigt profile function. Because of its convolution integral representation, the Voigt profile can be interpreted as the probability density function of the sum of two independent random variables with Gaussian density (due to the Doppler effect) and Lorentzian density (due to the pressure effect). Since these densities belong to the class of symmetric L\'evy stable distributions, a probabilistic generalization is proposed as the convolution of two arbitrary symmetric L\'evy densities. We study the case when the widths of the considered distributions depend on a scale-factor $\tau$ that is representative of spatial inhomogeneity or temporal non-stationarity. The evolution equations for this probabilistic generalization of the Voigt function are here introduced and interpreted as generalized diffusion equations containing two Riesz space-fractional derivatives, thus classified as space-fractional diffusion equations of double order. http://arxiv.org/abs/0711.4246 --------------------------------------------------------------- 7060. NON-MARKOVIAN DIFFUSION EQUATIONS AND PROCESSES: ANALYSIS AND SIMULATIONS Antonio Mura and Murad S. Taqqu and Francesco Mainardi In this paper we introduce and analyze a class of diffusion type equations related to certain non-Markovian stochastic processes. We start from the forward drift equation which is made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation can be interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time-evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the memory kernel K(t). We develop several applications and derive the exact solutions. We consider different stochastic models for the given equations providing path simulations. http://arxiv.org/abs/0712.0240 --------------------------------------------------------------- 7061. DISJOINTNESS OF REPRESENTATIONS ARISING IN HARMONIC ANALYSIS ON THE INFINITE-DIMENSIONAL UNITARY GROUP Vadim Gorin We prove pairwise disjointness of representations T_{z,w} of the infinite-dimensional unitary group. These representations provide a natural generalization of the regular representation for the case of "big" group U(\infty). They were introduced and studied by G.Olshanski and A.Borodin. Disjointness of the representations can be reduced to disjointness of certain probability measures on the space of paths in the Gelfand-Tsetlin graph. We prove the latter disjointness using probabilistic and combinatorial methods. http://arxiv.org/abs/0805.2660 --------------------------------------------------------------- 7062. ON THE DISTRIBUTION OF THE NODAL SETS OF RANDOM SPHERICAL HARMONICS Igor Wigman We study the length of the nodal set of eigenfunctions of the Laplacian on the $\spheredim$-dimensional sphere. It is well known that the eigenspaces corresponding to $\eigval=n(n+\spheredim-1)$ are the spaces $\eigspc$ of spherical harmonics of degree $n$, of dimension $\eigspcdim$. We use the multiplicity of the eigenvalues to endow $\eigspc$ with the Gaussian probability measure and study the distribution of the $\spheredim$- dimensional volume of the nodal sets of a randomly chosen function. The expected volume is proportional to $\sqrt{\eigval}$. One of our main results is bounding the variance of the volume to be $O(\frac{\eigval}{\sqrt{\eigspcdim}})$. In addition to the volume of the nodal set, we study its Leray measure. For every $n$, the expected value of the Leray measure is $\frac{1} {\sqrt{2\pi}}$. We are able to determine that the asymptotic form of the variance is $\frac{const}{\eigspcdim}$. http://arxiv.org/abs/0805.2768 --------------------------------------------------------------- 7063. GENERALIZED BACKWARD STOCHASTIC DIFFERENTIAL EQUATION WITH TWO REFLECTING BARRIERS AND STOCHASTIC QUADRATIC GROWTH E. H. Essaky and M. Hassani In this paper we study one-dimensional generalized reflected backward stochastic differential equation with two barriers and stochastic quadratic growth. We prove the existence of a maximal solution when there exists a semimartingale between the barriers L and U, the generator f is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z and without assuming any P-integrability conditions on the data. The proof of our result is based on the use of a comparison theorem, an exponential transformation and an approximation technique. Our result is applied to the Dynkin game problem as well as to the American game option. http://arxiv.org/abs/0805.2979 --------------------------------------------------------------- 7064. WIGNER FUNCTIONS AND STOCHASTICALLY PERTURBED LATTICE DYNAMICS Giada Basile (WIAS) and Stefano Olla (CEREMADE) and Herbert Spohn (D-Mutu-ZM) We consider lattice dynamics with a small stochastic perturbation of order \epsilon and prove that for a space-time scale of order \epsilon-1 the Wigner function evolves according to a linear transport equation describing inelastic collisions. For an energy and momentum conserving chain the transport equation predicts a slow decay, as 1/\sqrt{t}, for the energy current correlation in equilibrium. This is in agreement with previous studies using a different method. http://arxiv.org/abs/0805.3012 --------------------------------------------------------------- 7065. ASYMPTOTICS OF CHARACTERISTIC POLYNOMIALS OF WIGNER MATRICES AT THE EDGE OF THE SPECTRUM Holger K\"osters We investigate the asymptotic behaviour of the second-order correlation function of the characteristic polynomial of a Hermitian Wigner matrix at the edge of the spectrum. We show that the suitably rescaled second-order correlation function is asymptotically given by the Airy kernel, thereby generalizing the well-known result for the Gaussian Unitary Ensemble (GUE). Moreover, we obtain similar results for real-symmetric Wigner matrices. http://arxiv.org/abs/0805.3044 --------------------------------------------------------------- 7066. CONVERGENCE OF DEPENDENT WALKS IN A RANDOM SCENERY TO FBM-LOCAL TIME FRACTIONAL STABLE MOTIONS Serge Cohen (LSProba) and Cl\'ement Dombry (LMA) It is classical to approximate the distribution of fractional Brownian motion by a renormalized sum $ S_n $ of dependent Gaussian random variables. In this paper we consider such a walk $ Z_n $ that collects random rewards $ \xi_j $ for $ j \in \mathbb Z,$ when the ceiling of the walk $ S_n $ is located at $ j.$ The random reward (or scenery) $ \xi_j $ is independent of the walk and with heavy tail. We show the convergence of the sum of independent copies of $ Z_n$ suitably renormalized to a stable motion with integral representation, whose kernel is the local time of a fractional Brownian motion (fBm). This work extends a previous work where the random walk $ S_n$ had independent increments limits. http://arxiv.org/abs/0805.3054 --------------------------------------------------------------- 7067. POISSON-DIRICHLET DISTRIBUTION WITH SMALL MUTATION RATE Shui Feng The behavior of the Poisson-Dirichlet distribution with small mutation rate is studied through large deviations. The structure of the rate function indicates that the number of alleles is finite at the instant when mutation appears. The large deviation results are then used to study the asymptotic behavior of the homozygosity, and the Poisson-Dirichlet distribution with symmetric selection. The latter shows that several alleles can coexist when selection intensity goes to infinity in a particular way as the mutation rate approaches zero. http://arxiv.org/abs/0805.3113 --------------------------------------------------------------- 7068. HOW T-CELLS USE LARGE DEVIATIONS TO RECOGNIZE FOREIGN ANTIGENS Natali Zint and Ellen Baake and Frank den Hollander A stochastic model for the activation of T-cells is analysed. T-cells are part of the immune system and recognize foreign antigens against a background of the body's own molecules. The model under consideration is a slight generalization of a model introduced by Van den Berg, Rand and Burroughs in 2001, and is capable of explaining how this recognition works on the basis of rare stochastic events. With the help of a refined large deviation theorem and numerical evaluation it is shown that, for a wide range of parameters, T-cells can distinguish reliably between foreign antigens and self-antigens. http://arxiv.org/abs/q-bio/0605016 --------------------------------------------------------------- 7069. RANDOM MATRICES: A GENERAL APPROACH FOR THE LEAST SINGULAR VALUE PROBLEM Terence Tao and Van Vu Let $x$ be a complex random variable with mean zero and bounded variance. Let $N_{n}$ be the random matrix of size $n$ whose entries are iid copies of $x$ and $M$ be an arbitrary matrix. We give a general estimate for the least singular value of the matrix $M_{n}:=M + N_{n}$. In various special cases, our estimate extends or refines previous known results. http://arxiv.org/abs/0805.3167 --------------------------------------------------------------- 7070. BROWNIAN ENTROPIC REPULSION Itai Benjamini and Nathanael Berestycki We consider one-dimensional Brownian motion conditioned (in a suitable sense) to have a local time at every point and at every moment bounded by some fixed constant. Our main result shows that a phenomenon of entropic repulsion occurs: that is, this process is ballistic and has an asymptotic velocity approximately 4.58... as high as required by the conditioning (the exact value of this constant involves the first zero of a Bessel function). We also study the random walk case and show that the process is asymptotically ballistic but with an unknown speed. http://arxiv.org/abs/0805.3326 --------------------------------------------------------------- 7071. ON TIGHTNESS OF MUTUAL DEPENDENCE UPPERBOUND FOR SECRET-KEY CAPACITY OF MULTIPLE TERMINALS Chung Chan Csiszar and Narayan[3] show that the secret-key capacity with unlimited public discussion and the smallest achievable rate of communication for omniscience of a group of at least two active users sum up to the entropy rate of the discrete multiple memoryless sources for all terminals. They then derive a heuristically appealing upperbound[3,(26)] on the secret-key capacity, which is in the form of the information divergence from joint to product probability measure commonly interpreted as the mutual dependence of a set of random variables. Tightness of this bound would confirm its heuristic interpretation with the operational meaning of the secret-key capacity, i.e. the maximum mutual consensus among the active users that need not be explicitly described in public. While one can easily check that the bound is tight for any system with three or less users, testing the case with more users quickly becomes unmanageable. Yet, there is no apparent reason, other than its heuristic interpretation, that the bound is tight, nor is there a counter- example that suggests otherwise. This paper proves that the bound is indeed tight when all users are active, as a consequence of the polymatroidal structure[6] underlying the source coding problem. This already confirms the heuristic interpretation of the bound as a measure of mutual dependence of random variables. For the other case when some users are helpers, there is a counter-example with three active users and three helpers for which the bound is loose. http://arxiv.org/abs/0805.3200 --------------------------------------------------------------- 7072. ESTIMATION IN MODELS DRIVEN BY FRACTIONAL BROWNIAN MOTION Corinne Berzin and Jos\'e R. Le\'on Let $\{b_H(t),t\in\mathbb{R}\}$ be the fractional Brownian motion with parameter $0 0$, we use the orthogonal polynomials techniques developed by R. Killip and I. Nenciu to study certain linear statistics associated with the circular and Jacobi $\beta$ ensembles. We identify the distribution of these statistics then prove a joint central limit theorem. In the circular case, similar statements have been proved using different methods by a number of authors. In the Jacobi case these results are new. http://arxiv.org/abs/0805.3516 --------------------------------------------------------------- 7078. BERNSTEIN MEASURES ON CONVEX POLYTOPES Tatsuya Tate We define the notion of Bernstein measures and Bernstein approximations over general convex polytopes. This generalizes well-known Bernstein polynomials which are used to prove the Weierstrass approximation theorem on one dimensional intervals. We discuss some properties of Bernstein measures and approximations, and prove an asymptotic expansion of the Bernstein approximations for smooth functions which is a generalization of the asymptotic expansion of the Bernstein polynomials on the standard $m$-simplex obtained by Abel-Ivan and H\"{o}rmander. These are different from the Bergman- Bernstein approximations over Delzant polytopes recently introduced by Zelditch. We discuss relations between Bernstein approximations defined in this paper and Zelditch's Bergman-Bernstein approximations. http://arxiv.org/abs/0805.3379 --------------------------------------------------------------- 7079. RANDOM WALKS IN SPACE TIME MIXING ENVIRONMENTS Jean Bricmont and Antti Kupiainen We prove that random walks in random environments, that are exponentially mixing in space and time, are almost surely diffusive, in the sense that their scaling limit is given by the Wiener measure. http://arxiv.org/abs/0805.3455 --------------------------------------------------------------- 7080. PROBABILISTIC STUDY OF THE SPEED OF APPROACH TO EQUILIBRIUM FOR AN INELASTIC KAC MODEL Federico Bassetti and Lucia Ladelli and Eugenio Regazzini This paper deals with a one--dimensional model for granular materials, which boils down to an inelastic version of the Kac kinetic equation, with inelasticity parameter $p>0$. In particular, the paper provides bounds for certain distances -- such as specific weighted $\chi$--distances and the Kolmogorov distance -- between the solution of that equation and the limit. It is assumed that the even part of the initial datum (which determines the asymptotic properties of the solution) belongs to the domain of normal attraction of a symmetric stable distribution with characteristic exponent $\a=2/(1+p)$. With such initial data, it turns out that the limit exists and is just the aforementioned stable distribution. A necessary condition for the relaxation to equilibrium is also proved. Some bounds are obtained without introducing any extra--condition. Sharper bounds, of an exponential type, are exhibited in the presence of additional assumptions concerning either the behaviour, near to the origin, of the initial characteristic function, or the behaviour, at infinity, of the initial probability distribution function. http://arxiv.org/abs/0805.3508 --------------------------------------------------------------- 7081. AVERAGES OF RATIOS OF CHARACTERISTIC POLYNOMIALS IN CIRCULAR BETA-ENSEMBLES AND SUPER-JACK POLYNOMIALS Sho Matsumoto We study the averages of ratios of characteristic polynomials over circular $\beta$-ensembles, where $\beta$ is a positive real number. Using Jack polynomial theory, we obtain three expressions for ratio averages. Two of them are given as sums of super-Jack polynomials and another one is given by a hyperdeterminant. As applications, we give duality relations for ratio averages between $\beta$ and $4/\beta$. http://arxiv.org/abs/0805.3573 --------------------------------------------------------------- 7082. ON THE CLUSTER SIZE DISTRIBUTION FOR PERCOLATION ON SOME GENERAL GRAPHS Antar Bandyopadhyay and Jeffrey Steif and Adam Timar We show that for any Cayley graph, the probability (at any $p$) that the cluster of the origin has size n decays at a well-defined exponential rate (possibly 0). For general graphs, we relate this rate being positive in the supercritical regime with the amenability/nonamenability of the underlying graph. http://arxiv.org/abs/0805.3620 --------------------------------------------------------------- 7083. EXPLICIT ERROR BOUNDS FOR LAZY REVERSIBLE MARKOV CHAIN MONTE CARLO Daniel Rudolf We prove explicit, i.e., non-asymptotic, error bounds for Markov Chain Monte Carlo methods, such as the Metropolis algorithm. The problem is to compute the expectation (or integral) of f with respect to a measure which can be given by a density with respect to another measure. A straight simulation of the desired distribution by a random number generator is in general not possible. Thus it is reasonable to use Markov chain sampling with a burn-in. We study such an algorithm and extend the analysis of Lovasz and Simonovits (1993) to obtain an explicit error bound. http://arxiv.org/abs/0805.3587 --------------------------------------------------------------- 7084. SUCCESS EXPONENT OF WIRETAPPER: A TRADEOFF BETWEEN SECRECY AND RELIABILITY Chung Chan Equivocation rate has been widely used as an information-theoretic measure of security after Shannon[10]. It simplifies problems by removing the effect of atypical behavior from the system. In [9], however, Merhav and Arikan considered the alternative of using guessing exponent to analyze the Shannon's cipher system. Because guessing exponent captures the atypical behavior, the strongest expressible notion of secrecy requires the more stringent condition that the size of the key, instead of its entropy rate, to be equal to the size of the message. The relationship between equivocation and guessing exponent are also investigated in [6][7] but it is unclear which is a better measure, and whether there is a unifying measure of security. Instead of using equivocation rate or guessing exponent, we study the wiretap channel in [2] using the success exponent, defined as the exponent of a wiretapper successfully learn the secret after making an exponential number of guesses to a sequential verifier that gives yes/no answer to each guess. By extending the coding scheme in [2][5] and the converse proof in [4] with the new Overlap Lemma 5.2, we obtain a tradeoff between secrecy and reliability expressed in terms of lower bounds on the error and success exponents of authorized and respectively unauthorized decoding of the transmitted messages. From this, we obtain an inner bound to the strongly achievable public, private and guessing rate triple for which the exponents are strictly positive. The closure of this region contains the region in Theorem 1 of [2] when we treat equivocation rate as the guessing rate. It would be surprising if one can show that the subset relationship is strict, the region is tight, or a better coding scheme exists to improve it. These problems remain open. http://arxiv.org/abs/0805.3605 --------------------------------------------------------------- 7085. MULTIPLICATIVE FUNCTIONAL FOR REFLECTED BROWNIAN MOTION VIA DETERMINISTIC ODE Krzysztof Burdzy and John M. Lee We prove that a sequence of semi-discrete approximations converges to a multiplicative functional for reflected Brownian motion, which intuitively represents the Lyapunov exponent for the corresponding stochastic flow. The method of proof is based on a study of the deterministic version of the problem and the excursion theory. http://arxiv.org/abs/0805.3740 --------------------------------------------------------------- 7086. GENERALIZED CHINESE RESTAURANT CONSTRUCTION OF EXCHANGEABLE GIBBS PARTITIONS AND RELATED RESULTS Annalisa Cerquetti By resorting to sequential constructions of exchangeable random partitions (Pitman, 2006), and exploiting some known facts about generalized Stirling numbers, we derive a generalized Chinese restaurant process construction of exchangeable Gibbs partitions of type $\alpha$ (Gnedin and Pitman, 2006). Our construction represents the natural theoretical probabilistic framework in which to embed some recent results about a Bayesian nonparametric treatment of estimation problems arising in genetic experiment under Gibbs, species sampling, models priors. http://arxiv.org/abs/0805.3853 --------------------------------------------------------------- 7087. EXCURSIONS AWAY FROM A REGULAR POINT FOR ONE-DIMENSIONAL SYMMETRIC LEVY PROCESSES WITHOUT GAUSSIAN PART Kouji Yano The characteristic measure of excursions away from a regular point is studied for a class of symmetric Levy processes without Gaussian part. It is proved that the harmonic transform of the killed process enjoys Feller property. The result is applied to prove extremeness and oscillatory entrance properties of the excursion measure. http://arxiv.org/abs/0805.3881 --------------------------------------------------------------- 7088. A SET-VALUED FRAMEWORK FOR BIRTH-AND-GROWTH PROCESS Giacomo Aletti and Enea G. Bongiorno and Vincenzo Capasso We propose a set-valued framework for the well-posedness of birth-and- growth process. Our birth-and-growth model is rigorously defined as a suitable combination, involving Minkowski sum and Aumann integral, of two very general set-valued processes representing nucleation and growth respectively. The simplicity of the used geometrical approach leads us to avoid problems arising by an analytical definition of the front growth such as boundary regularities. In this framework, growth is generally anisotropic and, according to a mesoscale point of view, it is not local, i.e. for a fixed time instant, growth is the same at each space point. http://arxiv.org/abs/0805.3912 --------------------------------------------------------------- 7089. OPTIMAL H\"OLDER INDEX FOR DENSITY STATES OF SUPERPROCESSES WITH (1 + \BETA)-BRANCHING MECHANISM Klaus Fleischmann and Leonid Mytnik and Vitali Wachtel For 0 < \alpha \leq 2, a super-\alpha-stable motion X in R^d with branching of index 1 + \beta in (1,2) is considered. If d < \alpha / \beta, a dichotomy for the density of states X_t at fixed times t > 0 holds: the density function is locally H\"older continuous if d = 1 and \alpha > 1 + \beta, but locally unbounded otherwise. Moreover, in the case of continuity, we determine the optimal H\"older index. http://arxiv.org/abs/0805.3914 --------------------------------------------------------------- 7090. OPTIMAL INVESTMENT STRATEGY TO MINIMIZE OCCUPATION TIME Erhan Bayraktar and Virginia R. Young We find the optimal investment strategy to minimize the expected time that an individual's wealth stays below zero, the so-called {\it occupation time}. The individual consumes at a constant rate and invests in a Black-Scholes financial market consisting of one riskless and one risky asset, with the risky asset's price process following a geometric Brownian motion. We also consider an extension of this problem by penalizing the occupation time for the degree to which wealth is negative. http://arxiv.org/abs/0805.3981 --------------------------------------------------------------- 7091. ON THE UNIQUENESS OF THE INFINITE CLUSTER OF THE VACANT SET OF RANDOM INTERLACEMENTS A. Q. Teixeira We consider the model of random interlacements on Z^d introduced in [8]. For this model, we prove the uniqueness of the infinite component of the vacant set. As a consequence, we derive the continuity in u of the probability that the origin belongs to the infinite component of the vacant set at level u in the supercritical phase u < u_*. http://arxiv.org/abs/0805.4106 --------------------------------------------------------------- 7092. CONVERGENCE OF POINT PROCESSES WITH WEAKLY DEPENDENT POINTS Raluca Balan and Sana Louhichi For each $n \geq 1$, let $\{X_{j,n}\}_{1 \leq j \leq n}$ be a sequence of strictly stationary random variables. In this article, we give some asymptotic weak dependence conditions for the convergence in distribution of the point process $N_n=\sum_{j=1}^{n}\delta_{X_{j,n}}$ to an infinitely divisible point process. From the point process convergence, we obtain the convergence in distribution of the partial sum sequence $S_n=\sum_{j=1}^{n}X_{j,n}$ to an infinitely divisible random variable, whose L\'{e}vy measure is related to the canonical measure of the limiting point process. As examples, we discuss the case of triangular arrays which possess known (row-wise) dependence structures, like the strong mixing property, the association, or the dependence structure of a stochastic volatility model. http://arxiv.org/abs/0805.4128 --------------------------------------------------------------- 7093. ON THE DEPENDENCE STRUCTURE OF WAVELET COEFFICIENTS FOR SPHERICAL RANDOM FIELDS Xiaohong Lan and Domenico Marinucci We consider the correlation structure of the random coefficients for a wide class of wavelet systems on the sphere which was recently introduced in the literature. We provide necessary and sufficient conditions for these coefficients to be asymptotic uncorrelated in the real and in the frequency domain. Here, the asymptotic theory is developed in the high resolution sense. Statistical applications are also discussed, in particular with reference to the analysis of cosmological data. http://arxiv.org/abs/0805.4154 --------------------------------------------------------------- 7094. CONTINUOUS TIME RANDOM WALK AND PARAMETRIC SUBORDINATION IN FRACTIONAL DIFFUSION Rudolf Gorenflo and Francesco Mainardi and Alessandro Vivoli The well-scaled transition to the diffusion limit in the framework of the theory of continuous-time random walk (CTRW)is presented starting from its representation as an infinite series that points out the subordinated character of the CTRW itself. We treat the CTRW as a combination of a random walk on the axis of physical time with a random walk in space, both walks happening in discrete operational time. In the continuum limit we obtain a generally non-Markovian diffusion process governed by a space-time fractional diffusion equation. The essential assumption is that the probabilities for waiting times and jump-widths behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. By what we call parametric subordination, applied to a combination of a Markov process with a positively oriented L\'evy process, we generate and display sample paths for some special cases. http://arxiv.org/abs/cond-mat/0701126 --------------------------------------------------------------- 7095. FRACTIONAL DIFFUSION PROCESSES: PROBABILITY DISTRIBUTIONS AND CONTINUOUS TIME RANDOM WALK Rudolf Gorenflo and Francesco Mainardi A physical-mathematical approach to anomalous diffusion may be based on fractional diffusion equations and related random walk models. The fundamental solutions of these equations can be interpreted as probability densities evolving in time of peculiar self-similar stochastic processes: an integral representation of these solutions is here presented. A more general approach to anomalous diffusion is known to be provided by the master equation for a continuous time random walk (CTRW). We show how this equation reduces to our fractional diffusion equation by a properly scaled passage to the limit of compressed waiting times and jump widths. Finally, we describe a method of simulation and display (via graphics) results of a few numerical case studies. http://arxiv.org/abs/0709.3990 --------------------------------------------------------------- 7096. GENERALIZED STIRLING PERMUTATIONS, FAMILIES OF INCREASING TREES AND URN MODELS Svante Janson and Markus Kuba and Alois Panholzer Bona [2007+] studied the distribution of ascents, plateaux and descents in the class of Stirling permutations, introduced by Gessel and Stanley [1978]. Recently, Janson [2008+] showed the connection between Stirling permutations and plane recursive trees and proved a joint normal law for the parameters considered by Bona. Here we will consider generalized Stirling permutations extending the earlier results of Bona and Janson, and relate them with certain families of generalized plane recursive trees, and also $(k+1)$-ary increasing trees. We also give two different bijections between certain families of increasing trees, which both give as a special case a bijection between ternary increasing trees and plane recursive trees. In order to describe the (asymptotic) behaviour of the parameters of interests, we study three (generalized) Polya urn models using various methods. http://arxiv.org/abs/0805.4084 --------------------------------------------------------------- 7097. ON THE ENTROPY AND LOG-CONCAVITY OF COMPOUND POISSON MEASURES Oliver Johnson and Ioannis Kontoyiannis and Mokshay Madiman Motivated, in part, by the desire to develop an information-theoretic foundation for compound Poisson approximation limit theorems (analogous to the corresponding developments for the central limit theorem and for simple Poisson approximation), this work examines sufficient conditions under which the compound Poisson distribution has maximal entropy within a natural class of probability measures on the nonnegative integers. We show that the natural analog of the Poisson maximum entropy property remains valid if the measures under consideration are log-concave, but that it fails in general. A parallel maximum entropy result is established for the family of compound binomial measures. The proofs are largely based on ideas related to the semigroup approach introduced in recent work by Johnson for the Poisson family. Sufficient conditions are given for compound distributions to be log- concave, and specific examples are presented illustrating all the above results. http://arxiv.org/abs/0805.4112 --------------------------------------------------------------- 7098. CHAINS OF DISTRIBUTIONS, HIERARCHICAL BAYESIAN MODELS AND BENFORD'S LAW Dennis Jang and Jung Uk Kang and Alex Kruckman and Jun Kudo and Steven J. Miller Alex Ely Kossovsky recently conjectured that the distribution of leading digits of a chain of probability distributions converges to Benford's law as the length of the chain grows. We prove his conjecture in many cases, and provide an interpretation in terms of products of independent random variables and a central limit theorem. An important consequence is that in hierarchical Bayesian models priors tend to satisfy Benford's Law as the number of levels of the hyper-parameters increases. We give explicit formulas for the error terms as sums of Mellin transforms, which converges extremely rapidly as the number of terms in the chain grows. http://arxiv.org/abs/0805.4226 --------------------------------------------------------------- 7099. EXACT EDGEWORTH EXPANSION FOR A L\'{E}VY PROCESS Heikki J. Tikanm\"aki The one dimensional distribution of a L\'{e}vy process is not known in general even though its characteristic function is given by the famous L\'{e}vy-Khinchine theorem. This article gives an exact series representation for the one dimensional distribution of a L\'{e}vy process satisfying certain moment conditions. Moreover, this work clarifies an old result by Cram \'{e}r on Edgeworth expansions for the distribution function of a L\'{e}vy process. http://arxiv.org/abs/0805.4332 --------------------------------------------------------------- 7100. ON SUBEXPONENTIALITY OF THE L\'EVY MEASURE OF THE DIFFUSION INVERSE LOCAL TIME; WITH APPLICATIONS TO PENALIZATIONS Paavo Salminen and Pierre Vallois For a recurrent linear diffusion on $\R_+$ we study the asymptotics of the distribution of its local time at 0 as the time parameter tends to infinity. Under the assumption that the L\'evy measure of the inverse local time is subexponential this distribution behaves asymtotically as a multiple of the L\'evy measure. Using spectral representations we find the exact value of the multiple. For this we also need a result on the asymptotic behavior of the convolution of a subexponential distribution and an arbitrary distribution on $\R_+.$ The exact knowledge of the asymptotic behavior of the distribution of the local time allows us to analyze the process derived via a penalization procedure with the local time. This result generalizes the penalizations obtained in Roynette, Vallois and Yor \cite{rvyV} for Bessel processes. http://arxiv.org/abs/0805.4353 --------------------------------------------------------------- 7101. CONDITIONING ON AN EXTREME COMPONENT: MODEL CONSISTENCY AND REGULAR VARIATION ON CONES Bikramjit Das and Sidney I. Resnick Multivariate extreme value theory assumes a multivariate domain of attraction condition for the distribution of a random vector necessitating that each component satisfy a marginal domain of attraction condition. \cite{heffernan:tawn:2004} and \cite{heffernan:resnick:2007} developed an approximation to the joint distribution of the random vector by conditioning that one of the components be extreme. The prior papers left unresolved the consistency of different models obtained by conditioning on {different} components being extreme and we provide understanding of this issue. We also clarify the relationship between the conditional distributions and multivariate extreme value theory. We discuss conditions under which the two models are the same and when one can extend the conditional model to the extreme value model. We also discuss the relationship between the conditional extreme value model and standard regular variation on cones of the form $[0,\infty]\times(0,\infty]$ or $(0,\infty]\times[0,\infty]$. http://arxiv.org/abs/0805.4373 --------------------------------------------------------------- 7102. SPARSE POWER-EFFICIENT TOPOLOGIES FOR WIRELESS AD HOC SENSOR NETWORKS Amitabha Bagchi We study the problem of power-efficient routing for multihop wireless ad hoc sensor networks. The guiding insight of our work is that unlike an ad hoc wireless network, a wireless ad hoc sensor network does not require full connectivity among the nodes. As long as the sensing region is well covered by connected nodes, the network can perform its task. We consider two kinds of geometric random graphs as base interconnection structures: unit disk graphs $\UDG(2,\lambda)$ and $k$-nearest-neighbor graphs $\NN(2,k)$ built on points generated by a Poisson point process of density $\lambda$ in $\RR^2$. We provide subgraph constructions for these two models $\US(2,\lambda)$ and $\NS(2,k)$ and show that there are values $\lambda_s$ and $k_s$ above which these constructions have the following good properties: (i) they are sparse; (ii) they are power-efficient in the sense that the graph distance is no more than a constant times the Euclidean distance between any pair of points; (iii) they cover the space well; (iv) the subgraphs can be set up easily using local information at each node. We also describe a simple local algorithm for routing packets on these subgraphs. http://arxiv.org/abs/0805.4060 --------------------------------------------------------------- 7103. QUEUEING SYSTEM WITH PRE-SCHEDULED RANDOM ARRIVALS G. Guadagni and S. Ndreca and B. Scoppola We consider a point process obtained summing to each point $i$ of the set of the integer $\mathbb{Z}$ an i.i.d random variable $\xi_i$ having a variance that can be also much larger than 1. We compare the process obtained with this construction with the standard Poisson process, and we show that in some sense our process tends to converge for large variance of $\xi$ to the Poisson process in total variation. We then consider analytically and numerically a simple queueing system having our process as arrival process. This model is motivated by the study of air traffic systems. http://arxiv.org/abs/0805.4472 --------------------------------------------------------------- 7104. RANDOM WALKS ON DISCRETE CYLINDERS AND RANDOM INTERLACEMENTS Alain-Sol Sznitman We explore some of the connections between the local picture left by the trace of simple random walk on a discrete cylinder with base a d- dimensional torus, d at least 2, of side-length N running for times of order N^{2d} and the model of random interlacements recently introduced in arXiv:0704.2560. In particular we show that when the base becomes large, in the neighborhood of a point of the cylinder with a vertical component of order N^d, the complement of the set of points visited by the walk up to times of order N^{2d}, is close in distribution to the law of the vacant set of random interlacements at a level which is determined by an independent Brownian local time. The limit of the local pictures in the neighborhood of finitely many points is also derived. http://arxiv.org/abs/0805.4516 --------------------------------------------------------------- 7105. ON UPPER BOUNDS FOR THE TAIL DISTRIBUTION OF GEOMETRIC SUMS OF SUBEXPONENTIAL RANDOM VARIABLES Andrew Richards The approach used by Kalashnikov and Tsitsiashvili for constructing upper bounds for the tail distribution of a geometric sum with subexponential summands is reconsidered. By expressing the problem in a more probabilistic light, several improvements and one correction are made, which enables the constructed bound to be significantly tighter. Several examples are given, showing how to implement the theoretical result. http://arxiv.org/abs/0805.4548 --------------------------------------------------------------- 7106. ASYMPTOTIC PROPERTIES OF AN ESTIMATOR OF THE DRIFT COEFFICIENTS OF MULTIDIMENSIONAL ORNSTEIN-UHLENBECK PROCESSES THAT ARE NOT NECESSARILY STABLE Gopal K. Basak and Philip Lee In this paper, we investigate the consistency and asymptotic efficiency of an estimator of the drift matrix, $F$, of Ornstein-Uhlenbeck processes that are not necessarily stable. We consider all the cases. (1) The eigenvalues of $F$ are in the right half space (i.e., eigenvalues with positive real parts). In this case the process grows exponentially fast. (2) The eigenvalues of $F$ are on the left half space (i.e., the eigenvalues with negative or zero real parts). The process where all eigenvalues of $F$ have negative real parts is called a stable process and has a unique invariant (i.e., stationary) distribution. In this case the process does not grow. When the eigenvalues of $F$ have zero real parts (i.e., the case of zero eigenvalues and purely imaginary eigenvalues) the process grows polynomially fast. Considering (1) and (2) separately, we first show that an estimator, $\hat{F}$, of $F$ is consistent. We then combine them to present results for the general Ornstein-Uhlenbeck processes. We adopt similar procedure to show the asymptotic efficiency of the estimator. http://arxiv.org/abs/0805.4535 --------------------------------------------------------------- 7107. ON THE FIRST PASSAGE TIME FOR BROWNIAN MOTION SUBORDINATED BY A LEVY PROCESS T. R. Hurd and A. Kuznetsov This paper considers the class of L\'evy processes that can be written as a Brownian motion time changed by an independent L\'evy subordinator. Examples in this class include the variance gamma model, the normal inverse Gaussian model, and other processes popular in financial modeling. The question addressed is the precise relation between the standard first passage time and an alternative notion, which we call first passage of the second kind, as suggested by Hurd (2007) and others. We are able to prove that standard first passage time is the almost sure limit of iterations of first passage of the second kind. Many different problems arising in financial mathematics are posed as first passage problems, and motivated by this fact, we are lead to consider the implications of the approximation scheme for fast numerical methods for computing first passage. We find that the generic form of the iteration can be competitive with other numerical techniques. In the particular case of the VG model, the scheme can be further refined to give very fast algorithms. http://arxiv.org/abs/0805.4618 --------------------------------------------------------------- 7108. NUMERICAL COMPUTATIONS FOR BACKWARD DOUBLY SDES AND SPDES Yufeng Shi and Weiqiang Yang and Jing Yuan In this paper we present two numerical schemes of approximating solutions of backward doubly stochastic differential equations (BDSDEs for short). We give a method to discretize a BDSDE. And we also give the proof of the convergence of these two kinds of solutions for BDSDEs respectively. We give a sample of computation of BDSDEs. http://arxiv.org/abs/0805.4662 --------------------------------------------------------------- 7109. BACKWARD SDES WITH CONSTRAINED JUMPS AND QUASI-VARIATIONAL INEQUALITIES Idris Kharroubi (PMA and CREST) and Jin Ma and Huyen Pham (PMA and CREST) and Jianfeng Zhang We consider a class of backward stochastic differential equations (BSDEs) driven by Brownian motion and Poisson random measure, and subject to constraints on the jump component. We prove the existence and uniqueness of the minimal solution for the BSDEs by using a penalization approach. Moreover, we show that under mild conditions the minimal solutions to these constrained BSDEs can be characterized as the unique viscosity solution of quasi-variational inequalities (QVIs), which leads to a probabilistic representation for solutions to QVIs. Such a representation in particular gives a new stochastic formula for value functions of a class of impulse control problems. As a direct consequence we obtain a numerical scheme for the solution of such QVIs via the simulation of the penalized BSDEs. http://arxiv.org/abs/0805.4676 --------------------------------------------------------------- 7110. LES DEUX QUADRANGULATIONS INFINIES UNIFORMES ONT M\^EME LOI Laurent M\'enard We prove that the uniform infinite random quadrangulations introduced respectively by Chassaing-Durhuus and Krikun have the same distribution. http://arxiv.org/abs/0805.4687 --------------------------------------------------------------- 7111. DENSENESS OF CERTAIN SMOOTH L\'EVY FUNCTIONALS IN $\DD_{1,2}$ Christel Geiss and Eija Laukkarinen The Malliavin derivative for a L\'evy process $(X_t)$ can be defined on the space $\DD_{1,2}$ using a chaos expansion or in the case of a pure jump process also via an increment quotient operator \cite{sole-utzet-vives}. In this paper we define the Malliavin derivative operator $\D$ on the class $ \mathcal{S}$ of smooth random variables $f(X_{t_1}, ..., X_{t_n}),$ where $f$ is a smooth function with compact support. We show that the closure of $L_2(\Om) \supseteq \mathcal{S} \stackrel{\D}{\to} L_2(\m\otimes \mass)$ yields to the space $\DD_{1,2}.$ As an application we conclude that Lipschitz functions map from $\DD_{1,2}$ into $\DD_{1,2}.$ http://arxiv.org/abs/0805.4704 --------------------------------------------------------------- 7112. MARKOV CHAIN-BASED STABILITY ANALYSIS OF GROWING NETWORKS Zhenting Hou and Jinying Tong and Dinghua Shi From the perspective of probability, the stability of growing network is studied in the present paper. Using the DMS model as an example, we establish a relation between the growing network and Markov process. Based on the concept and technique of first-passage probability in Markov theory, we provide a rigorous proof for existence of the steady-state degree distribution, mathematically re-deriving the exact formula of the distribution. The approach based on Markov chain theory is universal and performs well in a large class of growing networks. http://arxiv.org/abs/0805.4765 --------------------------------------------------------------- 7113. DIFFERENTIABILITY OF STOCHASTIC FLOW OF REFLECTED BROWNIAN MOTIONS Krzysztof Burdzy We prove that a stochastic flow of reflected Brownian motions in a smooth multidimensional domain is differentiable with respect to its initial position. The derivative is a linear map represented by a multiplicative functional for reflected Brownian motion. The method of proof is based on excursion theory and analysis of the deterministic Skorokhod equation. http://arxiv.org/abs/0806.0119 --------------------------------------------------------------- 7114. MARKING (1,2) POINTS OF THE BROWNIAN WEB AND APPLICATIONS C. M. Newman (1) and K. Ravishankar (2) and E. Schertzer (1) ((1) Courant Inst. of Mathematical Sciences, NYU, (2) Dept. of Mathematics, SUNY College at New Paltz) The Brownian web (BW), which developed from the work of Arratia and then T\'{o}th and Werner, is a random collection of paths (with specified starting points) in one plus one dimensional space-time that arises as the scaling limit of the discrete web (DW) of coalescing simple random walks. Two recently introduced extensions of the BW, the Brownian net (BN) constructed by Sun and Swart, and the dynamical Brownian web (DyBW) proposed by Howitt and Warren, are (or should be) scaling limits of corresponding discrete extensions of the DW -- the discrete net (DN) and the dynamical discrete web (DyDW). These discrete extensions have a natural geometric structure in which the underlying Bernoulli left or right "arrow" structure of the DW is extended by means of branching (i.e., allowing left and right simultaneously) to construct the DN or by means of switching (i.e., from left to right and vice-versa) to construct the DyDW. In this paper we show that there is a similar structure in the continuum where arrow direction is replaced by the left or right parity of the (1,2) space-time points of the BW (points with one incoming path from the past and two outgoing paths to the future, only one of which is a continuation of the incoming path). We then provide a complete construction of the DyBW and an alternate construction of the BN to that of Sun and Swart by proving that the switching or branching can be implemented by a Poissonian marking of the (1,2) points. http://arxiv.org/abs/0806.0158 --------------------------------------------------------------- 7115. FROM BLACK-SCHOLES AND DUPIRE FORMULAE TO LAST PASSAGE TIMES OF LOCAL MARTINGALES. PART A : THE INFINITE TIME HORIZON Amel Bentata (PMA) and Marc Yor (PMA and Iuf) These notes are the first half of the contents of the course given by the second author at the Bachelier Seminar (February 8-15-22 2008) at IHP. They also correspond to topics studied by the first author for her Ph.D.thesis. http://arxiv.org/abs/0806.0239 --------------------------------------------------------------- 7116. BACKWARD STOCHASTIC PDES RELATED TO THE UTILITY MAXIMIZATION PROBLEM M. Mania and R. Tevzadze We study utility maximization problem for general utility functions using dynamic programming approach. We consider an incomplete financial market model, where the dynamics of asset prices are described by an $R^d$-valued continuous semimartingale. Under some regularity assumptions we derive backward stochastic partial differential equation (BSPDE) related directly to the primal problem and show that the strategy is optimal if and only if the corresponding wealth process satisfies a certain forward-SDE. As examples the cases of power, exponential and logarithmic utilities are considered. http://arxiv.org/abs/0806.0240 --------------------------------------------------------------- 7117. SUSCEPTIBILITY IN SUBCRITICAL RANDOM GRAPHS Svante Janson and Malwina J. Luczak We study the evolution of the susceptibility in the subcritical random graph $G(n,p)$ as $n$ tends to infinity. We obtain precise asymptotics of its expectation and variance, and show it obeys a law of large numbers. We also prove that the scaled fluctuations of the susceptibility around its deterministic limit converge to a Gaussian law. We further extend our results to higher moments of the component size of a random vertex, and prove that they are jointly asymptotically normal. http://arxiv.org/abs/0806.0252 --------------------------------------------------------------- 7118. PERTURBATIVE APPROACH ON FINANCIAL MARKETS Simone Scotti We study the point of transition between complete and incomplete financial models thanks to Dirichlet Forms methods. We apply recent techniques, developped by Bouleau, to hedging procedures in order to perturbate parameters and stochastic processes, in the case of a volatility parameter fixed but uncertain for traders; we call this model Perturbed Black Scholes (PBS) Model. We show that this model can reproduce at the same time a smile effect and a bid-ask spread; we exhibit the volatility function associated to the local-volatility model equivalent to PBS model when vanilla options are concerned. Lastly, we present a connection between Error Theory using Dirichlet Forms and Utility Function Theory. http://arxiv.org/abs/0806.0287 --------------------------------------------------------------- 7119. RISK PREMIUM IMPACT IN THE PERTURBATIVE BLACK SCHOLES MODEL Luca Regis and Simone Scotti We study the risk premium impact in the Perturbative Black Scholes model. The Perturbative Black Scholes model, developed by Scotti, is a subjective volatility model based on the classical Black Scholes one, where the volatility used by the trader is an estimation of the market one and contains measurement errors. In this article we analyze the correction to the pricing formulas due to the presence of an underlying drift different from the risk free return. We prove that, under some hypothesis on the parameters, if the asset price is a sub-martingale under historical probability, then the implied volatility presents a skewed structure, and the position of the minimum depends on the risk premium $\lambda$. http://arxiv.org/abs/0806.0307 --------------------------------------------------------------- 7120. THE GROWTH EXPONENT FOR PLANAR LOOP-ERASED RANDOM WALK Robert Masson We give a new proof of a result of Kenyon that the growth exponent for loop-erased random walks in two dimensions is 5/4. The proof uses the convergence of LERW to Schramm-Loewner evolution with parameter 2, and is valid for irreducible bounded symmetric random walks on any two-dimensional discrete lattice. http://arxiv.org/abs/0806.0357 --------------------------------------------------------------- 7121. DENSITY FLUCTUATIONS FOR A ZERO-RANGE PROCESS ON THE PERCOLATION CLUSTER Patricia Goncalves and Milton Jara We prove that the density fluctuations for a zero-range process evolving on the supercritical percolation cluster are given by a generalized Ornstein-Uhlenbeck process in the space of distributions $\mc S'(\bb R^d)$. http://arxiv.org/abs/0806.0362 --------------------------------------------------------------- 7122. COARSE GRAINING, FRACTIONAL MOMENTS AND THE CRITICAL SLOPE OF RANDOM COPOLYMERS F. Toninelli (Laboratoire de Physique and ENS Lyon and CNRS) For a much-studied model of random copolymer at a selective interface we prove that the slope of the critical curve in the weak-disorder limit is strictly smaller than 1, which is the value given by the annealed inequality. The proof is based on a coarse-graining procedure, combined with upper bounds on the fractional moments of the partition function. http://arxiv.org/abs/0806.0365 --------------------------------------------------------------- 7123. EVEN WALKS AND ESTIMATES OF HIGH MOMENTS OF LARGE WIGNER RANDOM MATRICES O. Khorunzhiy and V. Vengerovsky We revisit the problem of estimates of moments of random n-dimensional matrices of Wigner ensemble by using the approach elaborated by Ya. Sinai and A. Soshnikov and further developed by A. Ruzmaikina. Our main subject is given by the structure of closed even walks and their graphs that arise in these studies. We show that the total degree of a vertex of such a graph depends not only on the self-intersections degree of but also on the total number of all non-closed instants of self-intersections of the walk. This result is used to fill the gaps of earlier considerations. http://arxiv.org/abs/0806.0157 --------------------------------------------------------------- 7124. CONDITIONS FOR EXISTENCE AND SMOOTHNESS OF THE DISTRIBUTION DENSITY FOR AN ORNSTEIN-UHLENBECK PROCESS WITH LEVY NOISE Semen V.Bodnarchuk and Alexey M.Kulik Conditions are given, sufficient for the distribution of an Ornstein-Uhlenbeck process with L\'evy noise to be absolutely continuous or to possess a smooth density. For the processes with non-degenerate drift coefficient, these conditions are a necessary ones. A multidimensional analogue for the non-degeneracy condition on the drift coefficient is introduced. http://arxiv.org/abs/0806.0442 --------------------------------------------------------------- 7125. STABILITY OF THE LCD MODEL Zhenting Hou and Li Tan and Dinghua Shi In this paper, first-passage probability of Markov chains is used to get a strict proof of the existence of degree distribution of the LCD model presented by Bollobas (Random Structures and Algorithms 18(2001)). Also, a precise expression of degree distribution is presented. http://arxiv.org/abs/0806.0448 --------------------------------------------------------------- 7126. ON SUMS OF CONDITIONALLY INDEPENDENT SUBEXPONENTIAL RANDOM VARIABLES Serguei Foss and Andrew Richards The asymptotic tail-behaviour of sums of independent subexponential random variables is well understood, one of the main characteristics being \textit{the principle of the single big jump}. We study the case of dependent subexponential random variables, for both deterministic and random sums, using a fresh approach, by considering conditional independence structures on the random variables. We seek sufficient conditions for the results of the theory with independent random variables still to hold. For a subexponential distribution, we introduce the concept of a boundary class of functions, which we hope will be a useful tool in studying many aspects of subexponential random variables. The examples we give in the paper demonstrate a variety of effects owing to the dependence, and are also interesting in their own right. http://arxiv.org/abs/0806.0490 --------------------------------------------------------------- 7127. A DOUBLE PHASE TRANSITION ARISING FROM BROWNIAN ENTROPIC REPULSION Itai Benjamini and Nathanael Berestycki We analyze one-dimensional Brownian motion conditioned on a self- repelling behaviour. In the main result of this paper, it is shown that a double phase transition occurs when the growth of the local time at the origin is constrained (in a suitable way) to be slower than the function f(t)= \sqrt{t}(\log t)^{-c} at every time. In the subcritical phase (c<0), the process is recurrent and the local time at 0 is diffusive. In the intermediary phase (01), the process becomes transient. The proof exploits the Brownian entropic repulsion phenomenon. http://arxiv.org/abs/0806.0597 --------------------------------------------------------------- 7128. SUBEQUIVALENCE RELATIONS AND POSITIVE-DEFINITE FUNCTIONS A. Ioana and A.S. Kechris and and T. Tsankov We study a positive-definite function associated to a measure-preserving equivalence relation on a standard probability space and use it to measure quantitatively the proximity of subequivalence relations. This is combined with a recent co-inducing construction of Epstein to produce new kinds of mixing actions of an arbitrary infinite discrete group and it is also used to show that orbit equivalence of free, measure preserving, mixing actions of non-amenable groups is unclassifiable in a strong sense. Finally, in the case of property (T) groups we discuss connections with invariant percolation on Cayley graphs and the calculation of costs. http://arxiv.org/abs/0806.0430 --------------------------------------------------------------- 7129. BACKWARD UNIQUENESS AND THE EXISTENCE OF THE SPECTRAL LIMIT FOR SOME PARABOLIC SPDES Z. Brze\'zniak and M. Neklyudov The aim of this article is to study the asymptotic behaviour for large times of solutions to a certain class of stochastic partial differential equations of parabolic type. In particular, we will prove the backward uniqueness result and the existence of the spectral limit for abstract SPDEs and then show how these results can be applied to some concrete linear and nonlinear SPDEs. For example, we will consider linear parabolic SPDEs with gradient noise and stochastic NSEs with multiplicative noise. Our results generalize the results proved in Ghidaglia (1986) for deterministic PDEs. One of the difficulties with extending the results from Ghidaglia (1986) to the stochastic case is that the standard It\^o formula is not directly applicable to the case considered in this article. We use certain approximations to overcome this problem. Another difficulty is that conditions 1.3-1.4, p.779 in Ghidaglia (1986) have no natural counterpart in the stochastic case. We have only conditions (1.11), (1.12). As a result, we require rather strong assumptions on the regularity of solutions in the case of stochastic equations with quadratic nonlinearity. In the same time, this problem does not appear in the case of linear stochastic equations or if nonlinearity has no more than "linear growth". http://arxiv.org/abs/0806.0616 --------------------------------------------------------------- 7130. HOMOGENEOUS NUCLEATION FOR GLAUBER AND KAWASAKI DYNAMICS IN LARGE VOLUMES AT LOW TEMPERATURES Anton Bovier and Frank den Hollander and Cristian Spitoni In this paper we study metastability in large volumes at low temperatures. We consider both Ising spins subject to Glauber spin-flip dynamics and lattice gas particles subject to Kawasaki hopping dynamics. Let $\b$ denote the inverse temperature and let $\L_\b \subset \Z^2$ be a square box with periodic boundary conditions such that $\lim_{\b\to\infty}|\L_\b|=\infty$. We run the dynamics on $\L_\b$ starting from a random initial configuration where all the droplets (= clusters of plus-spins, respectively, clusters of particles)are small. For large $\b$, and for interaction parameters that correspond to the metastable regime, we investigate how the transition from the metastable state (with only small droplets) to the stable state (with one or more large droplets) takes place under the dynamics. This transition is triggered by the appearance of a single \emph{critical droplet} somewhere in $\L_\b$. Using potential- theoretic methods, we compute the \emph{average nucleation time} (= the first time a critical droplet appears and starts growing) up to a multiplicative factor that tends to one as $\b\to\infty$. It turns out that this time grows as $Ke^{\Gamma\b}/|\L_\b|$ for Glauber dynamics and $K\b e^{\Gamma\b}/|\L_ \b|$ for Kawasaki dynamics, where $\Gamma$ is the local canonical, respectively, grand-canonical energy to create a critical droplet and $K$ is a constant reflecting the geometry of the critical droplet, provided these times tend to infinity (which puts a growth restriction on $|\L_\b|$). The fact that the average nucleation time is inversely proportional to $|\L_\b|$ is referred to as \emph{homogeneous nucleation}, because it says that the critical droplet for the transition appears essentially independently in small boxes that partition $\L_\b$. http://arxiv.org/abs/0806.0755 --------------------------------------------------------------- 7131. ON SLOWDOWN AND SPEEDUP OF TRANSIENT RANDOM WALKS IN RANDOM ENVIRONMENT Alexander Fribergh and Nina Gantert and Serguei Popov We consider one-dimensional random walks in random environment which are transient to the right. Our main interest is in the study of the sub- ballistic regime, where at time n the particle is typically at a distance $O(n^ \kappa)$ from the origin, $\kappa\in(0,1)$. We investigate the probabilities of moderate deviations from this behavior. Specifically, we are interested in quenched and annealed probabilities of slowdown (at time n, the particle is in $O(n^{\nu_0})$, $\nu_0\in (0,\kappa)$), and speedup (at time n, the particle is around $n^{\nu_1}$, $\nu_1\in (\kappa,1)$), for the current location of the particle and for the hitting times. Also, we study probabilities of backtracking: at time n, the particle is located near $(-n^\nu)$, thus making an unusual excursion to the left. For the slowdown, our estimates are valid in the ballistic case as well. http://arxiv.org/abs/0806.0790 --------------------------------------------------------------- 7132. CONTINUOUS SPIN MEAN-FIELD MODELS: LIMITING KERNELS AND GIBBS PROPERTIES OF LOCAL TRANSFORMS C. Kuelske and A. A. Opoku We extend the notion of Gibbsianness for mean-field systems to the set- up of general (possibly continuous) local state spaces. We investigate the Gibbs properties of systems arising from an initial mean-field Gibbs measure by application of given local transition kernels. This generalizes previous case-studies made for spins taking finitely many values to the first step in direction to a general theory, containing the following parts: (1) A formula for the limiting conditional probability distributions of the transformed system. It holds both in the Gibbs and non-Gibbs regime and invokes a minimization problem for a "constrained rate-function". (2) A criterion for Gibbsianness of the transformed system for initial Lipschitz- Hamiltonians involving concentration properties of the transition kernels. (3) A continuity estimate for the single-site conditional distributions of the transformed system. While (2) and (3) have provable lattice-counterparts, the characterization of (1) is stronger in mean-field. As applications we show short-time Gibbsianness of rotator mean-field models on the (q-1)- dimensional sphere under diffusive time-evolution and the preservation of Gibbsianness under local coarse-graining of the initial local spin space. http://arxiv.org/abs/0806.0802 --------------------------------------------------------------- 7133. A CLASS OF NON HOMOGENEOUS SELF INTERACTING RANDOM PROCESSES WITH APPLICATIONS TO LEARNING IN GAMES AND VERTEX-REINFORCED RANDOM WALKS Michel Benaim (UNINE) and Olivier Raimond (LM-Orsay) Using an approximation by a set-valued dynamical system, this paper studies a class of non Markovian and non homogeneous stochastic processes on a finite state space. It provides an unified approach to simulated annealing type processes. It permits to study new models of vertex reinforced random walks and new models of learning in games including Markovian fictitious play. http://arxiv.org/abs/0806.0806 --------------------------------------------------------------- 7134. FLUCTUATION BOUNDS FOR THE ASYMMETRIC SIMPLE EXCLUSION PROCESS Marton Balazs and Timo Seppalainen We give a partly new proof of the fluctuation bounds for the second class particle and current in the stationary asymmetric simple exclusion process. One novelty is a coupling that preserves the ordering of second class particles in two systems that are themselves ordered coordinatewise. http://arxiv.org/abs/0806.0829 --------------------------------------------------------------- 7135. THE HEIGHT AND RANGE OF WATERMELONS WITHOUT WALL Thomas Feierl We determine the weak limit of the distribution of the random variables "height" and "range" on the set of p-watermelons without wall restriction as the number of steps tends to infinity. Additionally, we provide asymptotics for the moments of the random variable "height". http://arxiv.org/abs/0806.0037 --------------------------------------------------------------- 7136. ON RANDOMLY PLACED ARCS ON THE CIRCLE Arnaud Durand We completely describe in terms of Hausdorff measures the size of the set of points of the circle that are covered infinitely often by a sequence of random arcs with given lengths. We also show that this set is a set with large intersection. http://arxiv.org/abs/0806.0880 --------------------------------------------------------------- 7137. REFLECTED SOLUTIONS OF BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL EQUATIONS Weiqiang Yang and Yufeng Shi and Yangling Gu We study reflected solutions of one-dimensional backward doubly stochastic differential equations (BDSDEs in short). The ``reflected'' keeps the solution above a given stochastic process. We get the uniqueness and existence by penalization. For the existence of backward stochastic integral, our proof is different from [KKPPQ] slightly. We also obtain a comparison theorem for reflected BDSDEs. http://arxiv.org/abs/0806.0917 --------------------------------------------------------------- 7138. ASYMPTOTICS OF THE MAXIMAL RADIUS OF AN $L^R$-OPTIMAL SEQUENCE OF QUANTIZERS Gilles Pag\`es (PMA) and Abass Sagna (PMA) Let $P$ be a probability distribution on $\mathbb{R}^d$ (equipped with an Euclidean norm). Let $ r,s > 0 $ and assume $(\alpha_n)_{n \geq 1}$ is an (asymptotically) $L^r(P)$-optimal sequence of $n$-quantizers. In this paper we investigate the asymptotic behavior of the maximal radius sequence induced by the sequence $(\alpha_n)_{n \geq 1}$ and defined to be for every $n \geq 1$, $\ \rho(\alpha_n) = \max \{| a |, a \in \alpha_n \}$. We show that if ${\rm card(supp}(P))$ is infinite, the maximal radius sequence goes to $\sup \{| x |, x \in {\rm supp}(P) \}$ as $n$ goes to infinity. We then give the rate of convergence for two classes of distributions with unbounded support : distributions with exponential tails and distributions with polynomial tails. http://arxiv.org/abs/0806.0918 --------------------------------------------------------------- 7139. THE STOCHASTIC HAMILTON-JACOBI EQUATION Joan-Andreu L\'azaro-Cam\'i and Juan-Pablo Ortega We extend some aspects of the Hamilton-Jacobi theory to the category of stochastic Hamiltonian dynamical systems. More specifically, we show that the stochastic action satisfies the Hamilton-Jacobi equation when, as in the classical situation, it is written as a function of the configuration space using a regular Lagrangian submanifold. Additionally, we will use a variation of the Hamilton-Jacobi equation to characterize the generating functions of one-parameter groups of symplectomorphisms that allow to rewrite a given stochastic Hamiltonian system in a form whose solutions are very easy to find; this result recovers the classical solutions by reduction to the equilibrium of a standard Hamiltonian system. http://arxiv.org/abs/0806.0993 --------------------------------------------------------------- 7140. COMPETITION BETWEEN DISCRETE RANDOM VARIABLES, WITH APPLICATIONS TO OCCUPANCY PROBLEMS Julia Eaton and Anant Godbole and Betsy Sinclair Consider $n$ players whose "scores" are independent and identically distributed values $\{X_i\}_{i=1}^n$ from some discrete distribution $F $. We pay special attention to the cases where (i) $F$ is geometric with parameter $p\to0$ and (ii) $F$ is uniform on $\{1,2,...,N\}$; the latter case clearly corresponds to the classical occupancy problem. The quantities of interest to us are, first, the $U$-statistic $W$ which counts the number of "ties" between pairs $i,j$; second, the univariate statistic $Y_r$, which counts the number of strict $r$-way ties between contestants, i.e., episodes of the form ${X_i}_1={X_i}_2=...={X_i}_r$; $X_j\ne {X_i}_1;j\ne i_1,i_2,...,i_r$; and, last but not least, the multivariate vector $Z_{AB}=(Y_A,Y_{A+1},...,Y_B)$. We provide Poisson approximations for the distributions of $W$, $Y_r$ and $Z_{AB}$ under some general conditions. New results on the joint distribution of cell counts in the occupancy problem are derived as a corollary. http://arxiv.org/abs/0806.1007 --------------------------------------------------------------- 7141. SURVIVAL TIME OF RANDOM WALK IN RANDOM ENVIRONMENT AMONG SOFT OBSTACLES Nina Gantert and Serguei Popov and Marina Vachkovskaia We consider a Random Walk in Random Environment (RWRE) moving in an i.i.d. random field of obstacles. When the particle hits an obstacle, it disappears with a positive probability. We obtain quenched and annealed bounds on the tails of the survival time in the general $d$-dimensional case. We then consider a simplified one-dimensional model (where transition probabilities and obstacles are independent and the RWRE only moves to neighbour sites), and obtain finer results for the tail of the survival time. In addition, we study also the "mixed" probability measures (quenched with respect to the obstacles and annealed with respect to the transition probabilities and vice- versa) and give results for tails of the survival time with respect to these probability measures. Further, we apply the same methods to obtain bounds for the tails of the hitting times of Branching Random Walks in Random Environment (BRWRE). http://arxiv.org/abs/0806.1030 --------------------------------------------------------------- 7142. THE FRACTIONAL LANGEVIN EQUATION: BROWNIAN MOTION REVISITED Francesco Mainardi and Paolo Pironi We have revisited the Brownian motion on the basis of the fractional Langevin equation which turns out to be a particular case of the generalized Langevin equation introduced by Kubo on 1966. The importance of our approach is to model the Brownian motion more realistically than the usual one based on the classical Langevin equation, in that it takes into account also the retarding effects due to hydrodynamic backflow, i.e. the added mass and the Basset memory drag. On the basis of the two fluctuation-dissipation theorems and of the techniques of the Fractional Calculus we have provided the analytical expressions of the correlation functions (both for the random force and the particle velocity) and of the mean squared particle displacement. The random force has been shown to be represented by a superposition of the usual white noise with a "fractional" noise. The velocity correlation function is no longer expressed by a simple exponential but exhibits a slower decay, proportional to $t^{-3/2}$ as $t \to \infty$, which indeed is more realistic. Finally, the mean squared displacement has been shown to maintain, for sufficiently long times, the linear behaviour which is typical of normal diffusion, with the same diffusion coefficient of the classical case. However, the Basset history force induces a retarding effect in the establishing of the linear behaviour, which in some cases could appear as a manifestation of anomalous diffusion to be correctly interpreted in experimental measurements. http://arxiv.org/abs/0806.1010 --------------------------------------------------------------- 7143. NEGATIVE ENTROPY, PRESSURE AND ZERO TEMPERATURE: A L.D.P. FOR STATIONARY MARKOV CHAINS ON [0,1] Artur O. Lopes and Joana Mohr and Rafael R. Souza We analyze some properties of maximizing stationary Markov probabilities on the Bernoulli space $[0,1]^\mathbb{N}$, More precisely, we consider ergodic optimization for a continuous potential $A$, where $A: [0,1]^\mathbb{N} \to \mathbb{R}$ which depends only on the two first coordinates. We are interested in finding stationary Markov probabilities $\mu_\infty$ on $ [0,1]^ \mathbb{N}$ that maximize the value $ \int A d \mu,$ among all stationary Markov probabilities $\mu$ on $[0,1]^\mathbb{N}$. This problem correspond in Statistical Mechanics to the zero temperature case for the interaction described by the potential $A$. The main purpose of this paper is to show, under the hypothesis of uniqueness of the maximizing probability, a Large Deviation Principle for a family of absolutely continuous Markov probabilities $\mu_\beta$ which weakly converges to $\mu_\infty$. The probabilities $\mu_\beta$ are obtained via an information we get from a Perron operator and they satisfy a variational principle similar to the pressure. Under the hypothesis of $A$ being $C^2$ and the twist condition, that is, $\frac{\partial^2 A}{\partial_x \partial_y} (x,y) \neq 0$, for all $ (x,y) \in [0,1]^2$, we show the graph property. http://arxiv.org/abs/0806.1012 --------------------------------------------------------------- 7144. BREAKING THE CHAIN Michael Allman and Volker Betz We consider the motion of a Brownian particle in $\mathbb{R}$, moving between a particle fixed at the origin and another moving deterministically away at slow speed $\epsilon>0$. The middle particle interacts with its neighbours via a potential of finite range $b>0$, with a unique minimum at $a>0$, where $b<2a$. We say that the chain of particles breaks on the left- or right-hand side when the middle particle is greater than a distance $b$ from its left or right neighbour, respectively. We study the asymptotic location of the first break of the chain in the limit of small noise, in the case where $ \epsilon = \epsilon(\sigma)$ and $\sigma>0$ is the noise intensity. http://arxiv.org/abs/0806.1163 --------------------------------------------------------------- 7145. ASYMPTOTICS OF POSTERIORS FOR BINARY BRANCHING PROCESSES Didier Piau (IF) We compute the posterior distributions of the initial population and parameter of binary branching processes, in the limit of a large number of generations. We compare this Bayesian procedure with a more na\"ive one, based on hitting times of some random walks. In both cases, central limit theorems are available, with explicit variances. http://arxiv.org/abs/0806.1173 --------------------------------------------------------------- 7146. LIMIT THEOREMS FOR SAMPLE EIGENVALUES IN A GENERALIZED SPIKED POPULATION MODEL Zhidong Bai (KLASMOE and Dsap) and Jian-Feng Yao (IRMAR) In the spiked population model introduced by Johnstone (2001),the population covariance matrix has all its eigenvalues equal to unit except for a few fixed eigenvalues (spikes). The question is to quantify the effect of the perturbation caused by the spike eigenvalues. Baik and Silverstein (2006) establishes the almost sure limits of the extreme sample eigenvalues associated to the spike eigenvalues when the population and the sample sizes become large. In a recent work (Bai and Yao, 2008), we have provided the limiting distributions for these extreme sample eigenvalues. In this paper, we extend this theory to a {\em generalized} spiked population model where the base population covariance matrix is arbitrary, instead of the identity matrix as in Johnstone's case. New mathematical tools are introduced for establishing the almost sure convergence of the sample eigenvalues generated by the spikes. http://arxiv.org/abs/0806.1141 --------------------------------------------------------------- 7147. MONOTONE LOOP MODELS AND RATIONAL RESONANCE Alan Hammond and Richard Kenyon Let $T_{n,m}=\mathbb Z_n\times\mathbb Z_m$, and define a random mapping $\phi\colon T_{n,m}\to T_{n,m}$ by $\phi(x,y)=(x+1,y)$ or $(x,y+1)$ independently over $x$ and $y$ and with equal probability. We study the orbit structure of such ``quenched random walks'' $\phi$ in the limit $m,n\to \infty$, and show how it depends sensitively on the ratio $m/n$. For $m/n$ near a rational $p/q$, we show that there are likely to be on the order of $ \sqrt{n}$ cycles, each of length O(n), whereas for $m/n$ far from any rational with small denominator, there are a bounded number of cycles, and for typical $m/n $ each cycle has length on the order of $n^{4/3}$. http://arxiv.org/abs/0806.1236 --------------------------------------------------------------- 7148. SEMIGROUP INEQUALITIES, STOCHASTIC DOMINATION, HARDY'S INEQUALITY, AND STRONG ERGODICITY Carl Graham (CMAP) For the classical Lp spaces of signed measures on the integers, we devise a framework in which bounds for a sub-Markovian semigroup of interest can be obtained, up to a constant factor, from bounds for another tractable semigroup that dominates stochastically the first one. The main tools are the Hardy inequality, the definition of related auxiliary Lp spaces suited to take advantage of the domination, and the proof that the norms are equivalent to the classical ones if the reference measure is quasi-geometrically decreasing. We illustrate the results using birth-death and single-birth processes. http://arxiv.org/abs/0806.1263 --------------------------------------------------------------- 7149. COMPUTING EXPECTED TRANSITION EVENTS IN REDUCIBLE MARKOV CHAINS Brian D. Ewald and Jeffrey Humpherys and Jeremy West We present a method for computing the expected number of times certain transition events occur during the transient phase of a reducible Markov chain. Examples of events include time to absorption, number of visits to a state, traversals of a particular transition, loops from a state to itself, and arrivals to a state from a particular subset of states. http://arxiv.org/abs/0806.1291 --------------------------------------------------------------- 7150. KPZ IN ONE DIMENSIONAL RANDOM GEOMETRY OF MULTIPLICATIVE CASCADES Itai Benjamini and Oded Schramm We prove a formula relating the Hausdorff dimension of a subset of the unit interval and the Hausdorff dimension of the same set with respect to a random path matric on the interval, which is generated using a multiplicative cascade. When the random variables generating the cascade are exponentials of Gaussians, the well known KPZ formula of Knizhnik, Polyakov and Zamolodchikov from quantum gravity appears. This note was inspired by the recent work of Duplantier and Sheffield proving a somewhat different version of the KPZ formula for Liouville gravity. In contrast with the Liouville gravity setting, the one dimensional multiplicative cascade framework facilitates the determination of the Hausdorff dimension, rather than some expected box count dimension. http://arxiv.org/abs/0806.1347 --------------------------------------------------------------- 7151. THE ALEXANDER-ORBACH CONJECTURE HOLDS IN HIGH DIMENSIONS Gady Kozma and Asaf Nachmias We examine the incipient infinite cluster (IIC) of critical percolation in regimes where mean-field behavior have been established, namely when the dimension d is large enough or when d>6 and the lattice is sufficiently spread out. We find that random walk on the IIC exhibits anomalous diffusion with the spectral dimension d_s=4/3, that is, p_t(x,x)= t^{-2/3+o(1)}. This establishes a conjecture of Alexander and Orbach. En route we calculate the one-arm exponent with respect to the intrinsic distance. http://arxiv.org/abs/0806.1442 --------------------------------------------------------------- 7152. THE END OF SLEEPING BEAUTY'S NIGHTMARE Berry Groisman The way a rational agent changes her belief in certain propositions/hypotheses in the light of new evidence lies at the heart of Bayesian inference. The basic natural assumption, as summarized in van Fraassen's Reflection Principle ([1984]), would be that in the absence of new evidence the belief should not change. Yet, there are examples that are claimed to violate this assumption. The apparent paradox presented by such examples, if not settled, would demonstrate the inconsistency and/or incompleteness of the Bayesian approach and without eliminating this inconsistency, the approach cannot be regarded as scientific. The Sleeping Beauty Problem is just such an example. The existing attempts to solve the problem fall into three categories. The first two share the view that new evidence is absent, but differ about the conclusion of whether Sleeping Beauty should change her belief or not, and why. The third category is characterized by the view that, after all, new evidence (although hidden from the initial view) is involved. My solution is radically different and does not fall in either of these categories. I deflate the paradox by arguing that the two different degrees of belief presented in the Sleeping Beauty Problem are in fact beliefs in two different propositions, i.e. there is no need to explain the (un)change of belief. http://arxiv.org/abs/0806.1316 --------------------------------------------------------------- 7153. ON THE NUMBER OF MATRICES AND A RANDOM MATRIX WITH PRESCRIBED ROW AND COLUMN SUMS AND 0-1 ENTRIES Alexander Barvinok We consider the set Sigma(R,C) of all mxn matrices having 0-1 entries and prescribed row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n). We prove an asymptotic estimate for the cardinality |Sigma(R, C)| via the solution to a convex optimization problem. We show that if Sigma(R, C) is sufficiently large, then a random matrix D in Sigma(R, C) sampled from the uniform probability measure in Sigma(R,C) is close to a particular matrix Z=Z(R,C) that maximizes the entropy among all non-negative matrices with row sums R and column sums C. Similar results are obtained for 0-1 matrices with prescribed row and column sums and assigned zeros in some positions. http://arxiv.org/abs/0806.1480 --------------------------------------------------------------- 7154. IT\^O'S FORMULA FOR THE $L_{P}$-NORM OF STOCHASTIC $W^{1}_{P}$- VALUED PROCESSES N.V. Krylov We prove It\^o's formula for the $L_{p}$-norm of a stochastic $W^{1}_{p}$-valued processes appearing in the theory of SPDEs in divergence form. http://arxiv.org/abs/0806.1557 --------------------------------------------------------------- 7155. SLOW DECORRELATIONS IN KPZ GROWTH Patrik L. Ferrari (Weierstrass Institute and WIAS-Berlin) For stochastic growth models in the Kardar-Parisi-Zhang (KPZ) class in 1+1 dimensions, fluctuations grow as t^{1/3} during time t and the correlation length at a fixed time scales as t^{2/3}. In this note we discuss the scale of time correlations. It turns out that the space-time is non-trivially fibred, having slow directions having decorrelation exponent equal to 1 instead of the usual 2/3. These directions are the characteristic curves of the PDE associated to the surface's slope. As a consequence, previously proven results for space-like paths will hold in the whole space-time except along the slow curves. http://arxiv.org/abs/0806.1350 --------------------------------------------------------------- 7156. DISTAL ACTIONS AND SHIFTED CONVOLUTION PROPERTY C. R. E. Raja and R. Shah A locally compact group $G$ is said to have shifted convolution property (abbr. as SCP) if for every regular Borel probability measure $\mu$ on $G$, either $\sup_{x\in G} \mu ^n (Cx) \ra 0$ for all compact subsets $C$ of $G$, or there exist $x\in G$ and a compact subgroup $K$ normalised by $x$ such that $\mu^nx^{-n} \ra \omega_K$, the Haar measure on $K$. We first consider distality of factor actions of distal actions. It is shown that this holds in particular for factors under compact groups invariant under the action and for factors under the connected component of identity. We then characterize groups having SCP in terms of a readily verifiable condition on the conjugation action (point-wise distality). This has some interesting corollaries to distality of certain actions and Choquet Deny measures which actually motivated SCP and point-wise distal groups. We also relate distality of actions on groups to that of the extensions on the space of probability measures. http://arxiv.org/abs/0806.1820 --------------------------------------------------------------- 7157. STOCHASTIC EQUATIONS WITH DELAY: OPTIMAL CONTROL VIA BSDES AND REGULAR SOLUTIONS OF HAMILTON-JACOBI-BELLMAN EQUATIONS Marco Fuhrman and Federica Masiero and Gianmario Tessitore We consider an Ito stochastic differential equation with delay, driven by brownian motion, whose solution, by an appropriate reformulation, defines a Markov process $X$ with values in a space of continuous functions $ \mathbf C$, with generator $\mathcal L$. We then consider a backward stochastic differential equation depending on $X$, with unknown processes $(Y,Z) $, and we study properties of the resulting system, in particular we identify the process $Z$ as a deterministic functional of $X$. We next prove that the forward-backward system provides a suitable solution to a class of parabolic partial differential equations on the space $\mathbf C$ driven by $ \mathcal L$, and we apply this result to prove a characterization of the fair price and the hedging strategy for a financial market with memory effects. We also include applications to optimal stochastic control of differential equation with delay: in particular we characterize optimal controls as feedback laws in terms the process $X$. http://arxiv.org/abs/0806.1837 --------------------------------------------------------------- 7158. THE STOCHASTIC HEAT EQUATION DRIVEN BY A GAUSSIAN NOISE: GERM MARKOV PROPERTY Raluca Balan and Doyoon Kim Let $u=\{u(t,x);t \in [0,T], x \in {\mathbb{R}}^{d}\}$ be the process solution of the stochastic heat equation $u_{t}=\Delta u+ \dot F, u(0,\cdot)=0$ driven by a Gaussian noise $\dot F$, which is white in time and has spatial covariance induced by the kernel $f$. In this paper we prove that the process $u$ is locally germ Markov, if $f$ is the Bessel kernel of order $ \alpha=2k,k \in \bN_{+}$, or $f$ is the Riesz kernel of order $\alpha=4k,k \in \bN_{+}$. http://arxiv.org/abs/0806.1898 --------------------------------------------------------------- 7159. THE MIXING TIME EVOLUTION OF GLAUBER DYNAMICS FOR THE MEAN-FIELD ISING MODEL Jian Ding and Eyal Lubetzky and Yuval Peres We consider Glauber dynamics for the Ising model on the complete graph on $n$ vertices, known as the Curie-Weiss model. It is well-known that the mixing-time in the high temperature regime ($\beta < 1$) has order $n\log n$, whereas the mixing-time in the case $\beta > 1$ is exponential in $n$. Recently, Levin, Luczak and Peres proved that for any fixed $\beta < 1$ there is cutoff at time $[2(1-\beta)]^{-1} n\log n$ with a window of order $n$, whereas the mixing-time at the critical temperature $\beta=1$ is $\Theta(n^{3/2})$. It is natural to ask how the mixing-time transitions from $\Theta(n\log n)$ to $ \Theta(n^{3/2})$ and finally to $\exp(\Theta(n))$. That is, how does the mixing-time behave when $\beta=\beta(n)$ is allowed to tend to 1 as $n\to\infty$. In this work, we obtain a complete characterization of the mixing- time of the dynamics as a function of the temperature, as it approaches its critical point $\beta_c=1$. In particular, we find a scaling window of order $1/\sqrt{n}$ around the critical temperature. In the high temperature regime, $ \beta = 1 - \delta$ for some $0 < \delta < 1$ so that $\delta^2 n \to\infty$ with $n$, the mixing-time has order $(n/\delta)\log(\delta^2 n)$, and exhibits cutoff with constant 1/2 and window size $n/\delta$. In the critical window, $ \beta = 1\pm \delta$ where $\delta^2 n$ is O(1), there is no cutoff, and the mixing- time has order $n^{3/2}$. At low temperature, $\beta = 1 + \delta$ for $\delta > 0$ with $\delta^2 n \to\infty$ and $\delta=o(1)$, there is no cutoff, and the mixing time has order $(n/\delta)\exp(({3/4}+o(1))\delta^2 n)$. http://arxiv.org/abs/0806.1906 --------------------------------------------------------------- 7160. QUASI-STATIONARY RANDOM OVERLAP STRUCTURES AND THE CONTINUOUS CASCADES Jason Miller A random overlap structure (ROSt) is a measure on pairs (X,Q) where X is a locally finite sequence in the real line with a maximum and Q a positive semidefinite matrix of overlaps intrinsic to the particles X. Such a measure is said to be quasi-stationary provided that the joint law of the gaps of X and overlaps Q is stable under a stochastic evolution driven by a Gaussian sequence with covariance Q. Aizenman et al. have shown that quasi-stationary ROSts serve as an important computational tool in the study of the Sherrington- Kirkpatrick (SK) spin-glass model from the perspective of cavity dynamics and the related ROSt variational principle for its free energy. In this framework, the Parisi solution is reflected in the ansatz that the overlap matrix exhibit a certain hierarchical structure. Aizenman et al. have posed the question of whether the ansatz could be explained by showing that the only ROSts that are quasi-stationary in a robust sense are given by a special class of hierarchical ROSts known as both the Ruelle Probability Cascades as well the GREM. Arguin and Aizenman have given an affirmative answer in the special case that the set of values S_Q taken on by the entries of Q is finite. We prove that this result holds even when |S_Q| is infinite provided that Q satisfies the technical condition that the closure of S_Q has no limit points from below. This is relevant to the understanding of the ground states of the SK model, as they satisfy |S_Q| = infinity. http://arxiv.org/abs/0806.1915 --------------------------------------------------------------- 7161. ON DIVERGENCE FORM SPDES WITH VMO COEFFICIENTS N.V. Krylov We present several results on solvability in Sobolev spaces $W^{1}_{p} $ of SPDEs in divergence form in the whole space. http://arxiv.org/abs/0806.1925 --------------------------------------------------------------- 7162. ON DIVERGENCE FORM SPDES WITH VMO COEFFICIENTS IN A HALF SPACE N.V. Krylov We extend several known results on solvability in the Sobolev spaces $W^{1}_{p}$, $p\in[2,\infty)$, of SPDEs in divergence form in $ \bR^{d}_{+}$ to equations having coefficients which are discontinuous in the space variable. http://arxiv.org/abs/0806.1963 --------------------------------------------------------------- 7163. STOCHASTIC CALCULUS FOR SYMMETRIC MARKOV PROCESSES Z.-Q. Chen and P. J. Fitzsimmons and K. Kuwae and T.-S. Zhang Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an It\^{o} formula for Dirichlet processes is obtained. http://arxiv.org/abs/0806.2044 --------------------------------------------------------------- 7164. SWITCHING GAMES OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS Ying Hu (IRMAR) and Shanjian Tang (SCHOOL of Mathematical Sciences) In this paper, we study the switching game of one-dimensional backward stochastic differential equations (BSDEs). This gives rise to a new type of multi-dimensional obliquely reflected BSDEs, which is a system of BSDEs reflected on the boundary of a special unbounded convex domain along an oblique direction. The existence of the adapted solution is obtained by the penalization method, the monotone convergence, and the a priori estimations. The uniqueness is obtained by a verification method (the first component of any adapted solution is shown to be the vector value of a switching problem for Reflected BSDEs). Finally, we show the existence of both the value and the saddle point for the switching game. More specifically, we prove that the value process of the switching game is given by the first component of the solution of the multi-dimensional obliquely reflected BSDEs and the saddle point can also be constructed using the latter. http://arxiv.org/abs/0806.2058 --------------------------------------------------------------- 7165. STEIN'S METHOD AND CHARACTERS OF COMPACT LIE GROUPS Jason Fulman Stein's method is used to study the trace of a random element from a compact Lie group or symmetric space. Central limit theorems are proved using very little information: character values on a single element and the decomposition of the square of the trace into irreducible components. This is illustrated for Lie groups of classical type and Dyson's circular ensembles. The approach in this paper will be useful for the study of higher dimensional characters, where normal approximations need not hold. http://arxiv.org/abs/0806.2168 --------------------------------------------------------------- 7166. THE ORIENTED SWAP PROCESS Omer Angel and Alexander Holroyd and Dan Romik Particles labelled $1,...,n$ are arranged initially in increasing order. Subsequently, each pair of neighbouring particles that is currently in increasing order swaps according to a Poisson process of rate 1. We analyze the asymptotic behaviour of this process as $n$ goes to infinity. We prove that the space-time trajectories of individual particles converge (when suitably scaled) to a certain family of random curves with two points of non- differentiability, and that the permutation matrix at a given time converges to a certain deterministic measure with absolutely continuous and singular parts. The absorbing state (where all particles are in decreasing order) is reached at time $(2+o(1))n$. The finishing times of individual particles converge to deterministic limits with fluctuations asymptotically governed by the Tracy-Widom distribution. http://arxiv.org/abs/0806.2222 --------------------------------------------------------------- 7167. TREE-VALUED RESAMPLING DYNAMICS: MARTINGALE PROBLEMS AND APPLICATIONS Andreas Greven and Peter Pfaffelhuber and Anita Winter The measure-valued Fleming-Viot process is a diffusion which models the evolution of allele frequencies in a multi-type population of constant size. In the neutral setting the Kingman coalescent is known to generate the genealogies of the "individuals" in the population at a fixed time. The goal of the present paper is to replace this static point of view on the genealogies by an analysis of the evolving of genealogies. We use well-posed martingale problems to construct the tree-valued resampling dynamics for both the finite population (tree-valued Moran dynamics) and the infinite population (tree-valued Fleming-Viot dynamics). We show that the tree-valued Moran dynamics converge in the limit of large populations to the tree-valued Fleming-Viot dynamics. In the long-term behavior we derive the Kingman coalescent measure tree as the equilibrium. As an application we study the evolution of the distribution of the length of the tree spanned by sequentially sampled "individuals". http://arxiv.org/abs/0806.2224 --------------------------------------------------------------- 7168. A CHANGE OF VARIABLE FORMULA FOR THE 2D FRACTIONAL BROWNIAN MOTION OF HURST INDEX BIGGER OR EQUAL TO 1/4 Ivan Nourdin (PMA) We prove a change of variable formula for the 2D fractional Brownian motion of index H bigger of equal to 1/4. For H strictly bigger than 1/4, our formula coincides with that obtained by using the rough paths theory. For H=1/4 (the more interesting case), there is an additional term that is a classical Wiener integral against an independent standard Brownian motion. http://arxiv.org/abs/0806.2248 --------------------------------------------------------------- 7169. STRONG ASYMMETRIC LIMIT OF THE QUASI-POTENTIAL OF THE BOUNDARY DRIVEN WEAKLY ASYMMETRIC EXCLUSION PROCESS Lorenzo Bertini and Davide Gabrielli and Claudio Landim We consider the weakly asymmetric exclusion process on a bounded interval with particles reservoirs at the endpoints. The hydrodynamic limit for the empirical density, obtained in the diffusive scaling, is given by the viscous Burgers equation with Dirichlet boundary conditions. In the case in which the bulk asymmetry is in the same direction as the drift due to the boundary reservoirs, we prove that the quasi-potential can be expressed in terms of the solution to a one-dimensional boundary value problem which has been introduced by Enaud and Derrida \cite{de}. We consider the strong asymmetric limit of the quasi-potential and recover the functional derived by Derrida, Lebowitz, and Speer \cite{DLS3} for the asymmetric exclusion process. http://arxiv.org/abs/0806.2296 --------------------------------------------------------------- 7170. LOCAL BOOTSTRAP PERCOLATION Janko Gravner and Alexander E. Holroyd We study a variant of bootstrap percolation in which growth is restricted to a single active cluster. Initially there is a single active site at the origin, while other sites of Z^2 are independently occupied with small probability p, otherwise empty. Subsequently, an empty site becomes active by contact with 2 or more active neighbors, and an occupied site becomes active if it has an active site within distance 2. We prove that the entire lattice becomes active with probability exp[alpha(p)/p], where alpha(p) is between -pi^2/9 + c sqrt p and pi^2/9 + C sqrt p (-log p)^3. This corrects previous numerical predictions for the scaling of the correction term. http://arxiv.org/abs/0806.2313 --------------------------------------------------------------- 7171. SPECIAL POINTS OF THE BROWNIAN NET Emmanuel Schertzer and Rongfeng Sun and and Jan M. Swart The Brownian net, which has recently been introduced by Sun and Swart [SS08], and independently by Newman, Ravishankar and Schertzer [NRS08], generalizes the Brownian web by allowing branching. In this paper, we study the structure of the Brownian net in more detail. In particular, we give an almost sure classification of each point in $R^2$ according to the configuration of the Brownian net paths entering and leaving the point. Along the way, we establish various other structural properties of the Brownian net. http://arxiv.org/abs/0806.2326 --------------------------------------------------------------- 7172. A SHARP UNIFORM BOUND FOR THE DISTRIBUTION OF A SUM OF BERNOULLI RANDOM VARIABLES Roberto Cominetti and Jose Vaisman We establish a uniform bound for the distribution of a sum $S^n=X_1+... +X_n$ of independent non-homogeneous Bernoulli random variables with $P(X_i=1)=p_i$. Specifically, we prove that $\sigma^n P(S^n=i)\leq M$ where $\sigma^n$ denotes the standard deviation of $S^n$ and the constant $M~0.4688$ is the maximum of $u\mapsto\sqrt{2u} e^{-2u}\sum_{k=0}^\infty({u^k\over k!})^2$. http://arxiv.org/abs/0806.2350 --------------------------------------------------------------- 7173. RELATIONS BETWEEN INVASION PERCOLATION AND CRITICAL PERCOLATION IN TWO DIMENSIONS Michael Damron and Art\"em Sapozhnikov and B\'alint V\'agv\"olgyi We study invasion percolation in two dimensions. We compare connectivity properties of the origin's invaded region to those of (a) the critical percolation cluster of the origin and (b) the incipient infinite cluster. To exhibit similarities, we show that for any k > 0 the k-point function of the first pond has the same asymptotic behaviour as the probability that k points are in the critical cluster of the origin. More prominent, though, are the differences. We show that there are infinitely many ponds that contain many large disjoint p_c-open clusters. Further, for k > 1, we compute the exact decay rate of the distribution of the radius of the kth pond and see that it is strictly different than that of the radius of the critical cluster of the origin. We finish by showing that the invasion percolation measure and the incipient infinite cluster measure are mutually singular. http://arxiv.org/abs/0806.2425 --------------------------------------------------------------- 7174. UNIVERSAL STRUCTURES IN SOME MEAN FIELD SPIN GLASSES, AND AN APPLICATION Erwin Bolthausen and Nicola Kistler We discuss a spin glass reminiscent of the Random Energy Model, which allows in particular to recast the Parisi minimization into a more classical Gibbs variational principle, thereby shedding some light on the physical meaning of the order parameter of the Parisi theory. As an application, we study the impact of an extensive cavity field on Derrida's REM: Despite its simplicity, this model displays some interesting features such as ultrametricity and chaos in temperature. http://arxiv.org/abs/0806.2446 --------------------------------------------------------------- 7175. CENTRAL LIMIT THEOREMS FOR EIGENVALUES IN A SPIKED POPULATION MODEL Zhidong Bai and Jian-feng Yao In a spiked population model, the population covariance matrix has all its eigenvalues equal to units except for a few fixed eigenvalues (spikes). This model is proposed by Johnstone to cope with empirical findings on various data sets. The question is to quantify the effect of the perturbation caused by the spike eigenvalues. A recent work by Baik and Silverstein establishes the almost sure limits of the extreme sample eigenvalues associated to the spike eigenvalues when the population and the sample sizes become large. This paper establishes the limiting distributions of these extreme sample eigenvalues. As another important result of the paper, we provide a central limit theorem on random sesquilinear forms. http://arxiv.org/abs/0806.2503 --------------------------------------------------------------- 7176. QUENCHED LAW OF LARGE NUMBERS FOR BRANCHING BROWNIAN MOTION IN A RANDOM MEDIUM J\'anos Engl\"ander We study a spatial branching model, where the underlying motion is $d$-dimensional ($d\ge1$) Brownian motion and the branching rate is affected by a random collection of reproduction suppressing sets dubbed mild obstacles. The main result of this paper is the quenched law of large numbers for the population for all $d\ge1$. We also show that the branching Brownian motion with mild obstacles spreads less quickly than ordinary branching Brownian motion by giving an upper estimate on its speed. When the underlying motion is an arbitrary diffusion process, we obtain a dichotomy for the quenched local growth that is independent of the Poissonian intensity. More general offspring distributions (beyond the dyadic one considered in the main theorems) as well as mild obstacle models for superprocesses are also discussed. http://arxiv.org/abs/0806.2512 --------------------------------------------------------------- 7177. HOMOGENIZATION OF A SINGULAR RANDOM ONE-DIMENSIONAL PDE Bogdan Iftimie and \'Etienne Pardoux and Andrey Piatnitski This paper deals with the homogenization problem for a one-dimensional parabolic PDE with random stationary mixing coefficients in the presence of a large zero order term. We show that under a proper choice of the scaling factor for the said zero order terms, the family of solutions of the studied problem converges in law, and describe the limit process. It should be noted that the limit dynamics remain random. http://arxiv.org/abs/0806.2518 --------------------------------------------------------------- 7178. THE QUENCHED INVARIANCE PRINCIPLE FOR RANDOM WALKS IN RANDOM ENVIRONMENTS ADMITTING A BOUNDED CYCLE REPRESENTATION Jean-Dominique Deuschel and Holger K\"osters We derive a quenched invariance principle for random walks in random environments whose transition probabilities are defined in terms of weighted cycles of bounded length. To this end, we adapt the proof for random walks among random conductances by Sidoravicius and Sznitman (Probab. Theory Related Fields 129 (2004) 219--244) to the non-reversible setting. http://arxiv.org/abs/0806.2525 --------------------------------------------------------------- 7179. ON THE SMALL DEVIATION PROBLEM FOR SOME ITERATED PROCESSES Frank Aurzada and Mikhail Lifshits We derive general results on the small deviation behavior for some classes of iterated processes. This allows us, in particular, to calculate the rate of the small deviations for $n$-iterated Brownian motions and, more generally, for the iteration of $n$ fractional Brownian motions. We also give a new and correct proof of some results in E. Nane, Laws of the iterated logarithm for $\alpha$-time Brownian motion, Electron. J. Probab. 11 (2006), no. 18, 434--459. http://arxiv.org/abs/0806.2559 --------------------------------------------------------------- 7180. ON A CLASS OF OPTIMAL STOPPING PROBLEMS FOR DIFFUSIONS WITH DISCONTINUOUS COEFFICIENTS Ludger R\"uschendorf and Mikhail A. Urusov In this paper, we introduce a modification of the free boundary problem related to optimal stopping problems for diffusion processes. This modification allows the application of this PDE method in cases where the usual regularity assumptions on the coefficients and on the gain function are not satisfied. We apply this method to the optimal stopping of integral functionals with exponential discount of the form $E_x\int_0^{\tau}e^{-\lambda s}f(X_s) ds$, $\lambda\ge0$ for one-dimensional diffusions $X$. We prove a general verification theorem which justifies the modified version of the free boundary problem. In the case of no drift and discount, the free boundary problem allows to give a complete and explicit discussion of the stopping problem. http://arxiv.org/abs/0806.2561 --------------------------------------------------------------- 7181. OPTIMAL INVESTMENT AND CONSUMPTION IN A BLACK--SCHOLES MARKET WITH L\'EVY-DRIVEN STOCHASTIC COEFFICIENTS {\L}ukasz Delong and Claudia Kl\"uppelberg In this paper, we investigate an optimal investment and consumption problem for an investor who trades in a Black--Scholes financial market with stochastic coefficients driven by a non-Gaussian Ornstein--Uhlenbeck process. We assume that an agent makes investment and consumption decisions based on a power utility function. By applying the usual separation method in the variables, we are faced with the problem of solving a nonlinear (semilinear) first- order partial integro-differential equation. A candidate solution is derived via the Feynman--Kac representation. By using the properties of an operator defined in a suitable function space, we prove uniqueness and smoothness of the solution. Optimality is verified by applying a classical verification theorem. http://arxiv.org/abs/0806.2570 --------------------------------------------------------------- 7182. A UNIFIED FRAMEWORK FOR UTILITY MAXIMIZATION PROBLEMS: AN ORLICZ SPACE APPROACH Sara Biagini and Marco Frittelli We consider a stochastic financial incomplete market where the price processes are described by a vector-valued semimartingale that is possibly nonlocally bounded. We face the classical problem of utility maximization from terminal wealth, with utility functions that are finite-valued over $(a,\infty)$, $a\in\lbrack-\infty,\infty)$, and satisfy weak regularity assumptions. We adopt a class of trading strategies that allows for stochastic integrals that are not necessarily bounded from below. The embedding of the utility maximization problem in Orlicz spaces permits us to formulate the problem in a unified way for both the cases $a\in\mathbb{R}$ and $a=- \infty$. By duality methods, we prove the existence of solutions to the primal and dual problems and show that a singular component in the pricing functionals may also occur with utility functions finite on the entire real line. http://arxiv.org/abs/0806.2582 --------------------------------------------------------------- 7183. PALM DISTRIBUTIONS OF WAVE CHARACTERISTICS IN ENCOUNTERING SEAS Sofia Aberg and Igor Rychlik and M. Ross Leadbetter Distributions of wave characteristics of ocean waves, such as wave slope, waveheight or wavelength, are an important tool in a variety of oceanographic applications such as safety of ocean structures or in the study of ship stability, as will be the focus in this paper. We derive Palm distributions of several wave characteristics that can be related to steepness of waves for two different cases, namely for waves observed along a line at a fixed time point and for waves encountering a ship sailing on the ocean. The relation between the distributions obtained in the two cases is also given physical interpretation in terms of a ``Doppler shift'' that is related to the velocity of the ship and the velocities of the individual waves. http://arxiv.org/abs/0806.2718 --------------------------------------------------------------- 7184. CONDITIONALLY IDENTICALLY DISTRIBUTED SPECIES SAMPLING SEQUENCES Federico Bassetti and Irene Crimaldi and Fabrizio Leisen Conditional identity in distribution (Berti et al. (2004)) is a new type of dependence for random variables, which generalizes the well-known notion of exchangeability. In this paper, a class of random sequences, called Generalized Species Sampling Sequences, is defined and a condition to have conditional identity in distribution is given. Moreover, a class of generalized species sampling sequences that are conditionally identically distributed is introduced and studied: the Generalized Ottawa sequences (GOS). This class contains a '`randomly reinforced'' version of the P\'olya urn and of the Blackwell-MacQueen urn scheme. For the empirical means and the predictive means of a GOS, we prove two convergence results toward suitable mixtures of Gaussian distributions. The first one is in the sense of stable convergence and the second one in the sense of almost sure conditional convergence. In the last part of the paper we study the length of the partition induced by a GOS at time $n$, i.e. the random number of distinct values of a GOS until time $n $. Under suitable conditions, we prove a strong law of large numbers and a central limit theorem in the sense of stable convergence. All the given results in the paper are accompanied by some examples. http://arxiv.org/abs/0806.2724 --------------------------------------------------------------- 7185. ${L^P}$-VARIATIONS FOR MULTIFRACTAL FRACTIONAL RANDOM WALKS Carenne Lude\~na A multifractal random walk (MRW) is defined by a Brownian motion subordinated by a class of continuous multifractal random measures $M[0,t], 0\le t \le1$. In this paper we obtain an extension of this process, referred to as multifractal fractional random walk (MFRW), by considering the limit in distribution of a sequence of conditionally Gaussian processes. These conditional processes are defined as integrals with respect to fractional Brownian motion and convergence is seen to hold under certain conditions relating the self-similarity (Hurst) exponent of the fBm to the parameters defining the multifractal random measure $M$. As a result, a larger class of models is obtained, whose fine scale (scaling) structure is then analyzed in terms of the empirical structure functions. Implications for the analysis and inference of multifractal exponents from data, namely, confidence intervals, are also provided. http://arxiv.org/abs/0806.2731 --------------------------------------------------------------- 7186. A MIXED SINGULAR/SWITCHING CONTROL PROBLEM FOR A DIVIDEND POLICY WITH REVERSIBLE TECHNOLOGY INVESTMENT Vathana Ly Vath and Huy\^en Pham and St\'ephane Villeneuve We consider a mixed stochastic control problem that arises in Mathematical Finance literature with the study of interactions between dividend policy and investment. This problem combines features of both optimal switching and singular control. We prove that our mixed problem can be decoupled in two pure optimal stopping and singular control problems. Furthermore, we describe the form of the optimal strategy by means of viscosity solution techniques and smooth-fit properties on the corresponding system of variational inequalities. Our results are of a quasi-explicit nature. From a financial viewpoint, we characterize situations where a firm manager decides optimally to postpone dividend distribution in order to invest in a reversible growth opportunity corresponding to a modern technology. In this paper a reversible opportunity means that the firm may disinvest from the modern technology and return back to its old technology by receiving some gain compensation. The results of our analysis take qualitatively different forms depending on the parameters values. http://arxiv.org/abs/0806.2745 --------------------------------------------------------------- 7187. VARIANCE BOUNDING MARKOV CHAINS Gareth O. Roberts and Jeffrey S. Rosenthal We introduce a new property of Markov chains, called variance bounding. We prove that, for reversible chains at least, variance bounding is weaker than, but closely related to, geometric ergodicity. Furthermore, variance bounding is equivalent to the existence of usual central limit theorems for all $L^2$ functionals. Also, variance bounding (unlike geometric ergodicity) is preserved under the Peskun order. We close with some applications to Metropolis-- Hastings algorithms. http://arxiv.org/abs/0806.2747 --------------------------------------------------------------- 7188. STOCHASTIC IMPULSE CONTROL OF NON-MARKOVIAN PROCESSES Boualem Djehiche and Said Hamadene and Ibtissam Hdhiri We consider a class of stochastic impulse control problems of general stochastic processes i.e. not necessarily Markovian. Under fairly general conditions we establish existence of an optimal impulse control. We also prove existence of combined optimal stochastic and impulse control of a fairly general class of diffusions with random coefficients. Unlike, in the Markovian framework, we cannot apply quasi-variational inequalities techniques. We rather derive the main results using techniques involving reflected BSDEs and the Snell envelope. http://arxiv.org/abs/0806.2761 --------------------------------------------------------------- 7189. CENTRAL LIMIT THEOREM FOR SIGNAL-TO-INTERFERENCE RATIO OF REDUCED RANK LINEAR RECEIVER G. M. Pan and W. Zhou Let $\mathbf{s}_k=\frac{1}{\sqrt{N}}(v_{1k},...,v_{Nk})^T,$ with $\{v_{ik},i,k=1,...\}$ independent and identically distributed complex random variables. Write $\mathbf{S}_k=(\mathbf{s}_1,...,\mathbf {s}_{k-1},\mathbf{s}_{k+1},... ,\mathbf{s}_K),$ $\mathbf{P}_k= \operatorname {diag}(p_1,...,p_{k-1},p_{k+1},...,p_K)$, $\mathbf{R}_k=(\mathbf{S}_k\mathbf{P}_k\mathbf{S}_k^*+\sigma ^2\mathbf{I})$ and $\mathbf{A}_{km}=[\mathbf{s}_k,\mathbf{R}_k\mathbf{s}_k,... ,\mathbf{R}_k^{m-1}\mathbf{s}_k]$. Define $\beta_{km}=p_k\mathbf{s}_k^*\mathbf{A}_{km}(\mathbf {A}_{km}^*\times\ mathbf{R}_k\mathbf{A}_{km})^{-1}\mathbf{A}_{km}^*\mathbf{s}_k$, referred to as the signal-to-interference ratio (SIR) of user $k$ under the multistage Wiener (MSW) receiver in a wireless communication system. It is proved that the output SIR under the MSW and the mutual information statistic under the matched filter (MF) are both asymptotic Gaussian when $N/K\to c>0$. Moreover, we provide a central limit theorem for linear spectral statistics of eigenvalues and eigenvectors of sample covariance matrices, which is a supplement of Theorem 2 in Bai, Miao and Pan [Ann. Probab. 35 (2007) 1532--1572]. And we also improve Theorem 1.1 in Bai and Silverstein [Ann. Probab. 32 (2004) 553--605]. http://arxiv.org/abs/0806.2768 --------------------------------------------------------------- 7190. STATIONARY MAX-STABLE FIELDS ASSOCIATED TO NEGATIVE DEFINITE FUNCTIONS Zakhar Kabluchko and Martin Schlather and Laurens de Haan Let $W_i(\cdot)$, $i\in\NN$, be independent copies of a zero mean gaussian process $\{W(t), t\in\RR^d\}$ with stationary increments; and denote by $\sigma^2(t)$ the variance of $W(t)$. Independently from $W_i$, let $\sum_{i=1}^\infty \delta_{U_i}$ be a Poisson point process on the real axis with intensity $e^{-y}dy$. We show that the law of the random family of functions $\{V_i(\cdot), i\in\NN\}$ defined by $V_i(t)=U_i+W_i(t)-\sigma^2(t)/2$ is translation invariant. In particular, the process $\eta(t)=\bigvee_{i=1}^{\infty}V_i(t)$ is a stationary max- stable process with standard Gumbel margins. The process $\eta$ arises as a limit of a suitably normalized and rescaled pointwise maximum of $n$ i.i.d. stationary gaussian processes as $n\to\infty$ if and only if $W$ is a (non- isotropic) fractional Brownian motion on $\RR^d$. Under suitable conditions on $W $, the process $\eta$ has a mixed moving maxima representation. http://arxiv.org/abs/0806.2780 --------------------------------------------------------------- 7191. ANOTHER CORRECTION. ERROR ESTIMATES FOR BINOMIAL APPROXIMATIONS OF GAME OPTIONS Yan Dolinsky and Yuri Kifer The Annals of Applied Probability 16 (2006) 984--1033 [URL: http://projecteuclid.org/euclid.aoap/1151592257] http://arxiv.org/abs/0806.2782 --------------------------------------------------------------- 7192. ON THE ERGODICITY OF THE ADAPTIVE METROPOLIS ALGORITHM ON UNBOUNDED DOMAINS Eero Saksman and Matti Vihola This paper describes sufficient conditions to ensure the correct ergodicity of the Adaptive Metropolis (AM) algorithm of Haario, Saksman, and Tamminen (2001) [8], for target distributions with a non-compact support. The conditions ensuring a strong law of large numbers and a central limit theorem require that the tails of the target density decay super-exponentially, and have regular enough convex contours. The result is based on the ergodicity of an auxiliary process that is sequentially constrained to feasible adaptation sets, and independent estimates of the growth rate of the AM chain and the corresponding geometric drift constants. The ergodicity result of the constrained process is obtained through a modification of the approach due to Andrieu and Moulines (2006) [1]. http://arxiv.org/abs/0806.2933 --------------------------------------------------------------- 7193. NEW TECHNIQUES FOR EMPIRICAL PROCESS OF DEPENDENT DATA Herold Dehling and Olivier Durieu and Dalibor Voln\'y We present a new technique for proving empirical process invariance principle for stationary processes $(X_n)_{n\geq 0}$. The main novelty of our approach lies in the fact that we only require the central limit theorem and a moment bound for a restricted class of functions $(f(X_n))_{n\geq 0}$, not containing the indicator functions. Our approach can be applied to Markov chains and dynamical systems, using spectral properties of the transfer operator. Our proof consists of a novel application of chaining techniques. http://arxiv.org/abs/0806.2941 --------------------------------------------------------------- 7194. A FOURTH MOMENT INEQUALITY FOR FUNCTIONALS OF STATIONARY PROCESSES Olivier Durieu In this paper, a fourth moment bound for partial sums of functional of strongly ergodic Markov chain is established. This type of inequality plays an important role in the study of empirical process invariance principle. This one is specially adapted to the technique of Dehling, Durieu and Voln\'y (2008). The same moment bound can be proved for dynamical system whose transfer operator has some spectral properties. Examples of applications are given. http://arxiv.org/abs/0806.2980 --------------------------------------------------------------- 7195. THE HIGH TEMPERATURE ISING MODEL ON THE TRIANGULAR LATTICE IS A CRITICAL PERCOLATION MODEL Andras Balint and Federico Camia and Ronald Meester The Ising model at inverse temperature $\beta$ and zero external field can be obtained via the Fortuin-Kasteleyn (FK) random-cluster model with $q=2$ and density of open edges $p=1-e^{-\beta}$ by assigning spin +1 or -1 to each vertex in such a way that (1) all the vertices in the same FK cluster get the same spin and (2) +1 and -1 have equal probability. We generalize the above procedure by assigning spin +1 with probability $r$ and -1 with probability $1-r$, with $r \in [0,1]$, while keeping condition (1). For fixed $ \beta$, this generates a dependent (spin) percolation model with parameter $r$. We show that, on the triangular lattice and for $\beta<\beta_c$, this model has a percolation phase transition at $r=1/2$, corresponding to the Ising model. This sheds some light on the conjecture that the high temperature Ising model on the triangular lattice is in the percolation universality class and that its scaling limit can be described in terms of SLE$_6$. We also prove uniqueness of the infinite +1 cluster for $r>1/2$, sharpness of the percolation phase transition (by showing exponential decay of the cluster size distribution for $r<1/2$), and continuity of the percolation function for all $r \in [0,1]$. http://arxiv.org/abs/0806.3020 --------------------------------------------------------------- 7196. A NOTE ON DOMINANT CONTRACTIONS OF JORDAN ALGEBRAS Farrukh Mukhamedov and Seyit Temir and Hasan Akin In the paper we consider two positive contractions $T,S:L^{1}(A,\tau)\longrightarrow L^{1}(A,\tau)$ such that $T\leq S$, here $(A,\t)$ is a semi-finite $JBW$-algebra. If there is an $n_{0}\in \mathbb{N}$ such that $\|S^{n_{0}}-T^{n_{0}}\|<1$. Then we prove that $\|S^{n}- T^{n}\|<1$ holds for every $n\geq n_{0}.$ http://arxiv.org/abs/0806.2926 --------------------------------------------------------------- 7197. STOCHASTIC CONTROL UP TO A HITTING TIME: OPTIMALITY AND ROLLING- HORIZON IMPLEMENTATION Debasish Chatterjee and Eugenio Cinquemani and Giorgos Chaloulos and John Lygeros We present a dynamic programming-based solution to a stochastic optimal control problem up to a hitting time for a discrete-time Markov control process. Firstly, we determine an optimal control policy to steer the process toward a compact target set while simultaneously minimizing an expected discounted cost. We then provide a rolling-horizon strategy for approximating the optimal policy, together with quantitative characterization of its sub-optimality with respect to the optimal policy. Finally, we address related issues of asymptotic discount-optimality of the value-iteration policy. Both the state and action spaces are assumed to be Polish. http://arxiv.org/abs/0806.3008 --------------------------------------------------------------- 7198. LAWS OF THE ITERATED LOGARITHM FOR A CLASS OF ITERATED PROCESSES Erkan Nane Let $X=\{X_t, t\geq 0\}$ be a Brownian motion or a spectrally negative stable process of index $1<\a<2$. Let $E=\{E_t,t\geq 0\}$ be the hitting time of a stable subordinator of index $0<\beta<1$ independent of $X$. We use a connection between $X(E_t)$ and the stable subordinator of index $ \beta/\a$ to derive information on the path behavior of $X(E_t)$. This is an extension of the connection of iterated Brownian motion and (1/4)-stable subordinator due to Bertoin \cite{bertoin}. Using this connection, we obtain various laws of the iterated logarithm for $X(E_t)$. In particular, we establish law of the iterated logarithm for local time Brownian motion, $X(L_t)$, where $X$ is a Brownian motion (the case $\a=2$) and $L_t$ is the local time at zero of a stable process $Y$ of index $1<\a_2\leq 2$ independent of $X$. In this case $E_{\rho t}=L_t$ with $\beta=1-1/\a_2$ for some constant $\rho>0$. This establishes the lower bound in the law of the iterated logarithm which we could not prove with the techniques of our paper \cite{MNX}. We also obtain exact small balll probability for $X(E_t)$ using ideas from \cite{aurzada}. http://arxiv.org/abs/0806.3126 --------------------------------------------------------------- 7199. DIRECTIONALLY CONVEX ORDERING OF RANDOM MEASURES, SHOT NOISE FIELDS AND SOME APPLICATIONS TO WIRELESS COMMUNICATIONS Bartlomiej Blaszczyszyn (INRIA Rocquencourt) and Dhandapani Yogeshwaran Directionally convex ($dcx$) ordering is a tool for comparison of dependence structure of random vectors that also takes into account the variability of the marginal distributions. When extended to random fields it oncerns comparison of all finite dimensional distributions. Viewing locally finite measures as non-negative fields of measure-values indexed by the bounded Borel subsets of the space, in this paper we formulate and study the $dcx$ ordering of random measures on locally compact spaces. We show that the $dcx$ order is preserved under some of the natural operations considered on random measures and point processes, such as independent superposition and thinning. Further operations such as independent marking and displacement, though do not preserve the $dcx$ order on all point processes, are shown to preserve the order on Cox point processes. We also examine the impact of $dcx$ order on the second moment properties, in particular on clustering and on Palm distributions. Comparisons of Ripley's functions, pair correlation functions as well as examples seem to indicate that p.p. higher in $dcx$ order cluster more. As the main result, we show that non-negative integral (shot-noise) fields with respect to $dcx$ ordered random measures inherit this ordering from the measures. Numerous applications of this result are shown, in particular to comparison of various Cox processes and some performance measures of wireless networks, in both of which shot-noise fields appear as key ingredients. We also mention a few pertinent open questions. http://arxiv.org/abs/0806.3180 --------------------------------------------------------------- 7200. EXISTENCE OF A CRITICAL POINT FOR THE INFINITE DIVISIBILITY OF SQUARES OF GAUSSIAN VECTORS IN $R^{2}$ WITH NON--ZERO MEAN Michael B. Marcus and Jay Rosen Let $G=(G_{1},G_{2})$ be a Gaussian vector in $R^{2}$ with $EG_{1}G_{2}\neq 0$. Let $c_{1},c_{2}\in R^{1}$. A necessary and sufficient condition for $G=((G_{1}+c_{1}\alpha)^{2},(G_{2}+c_{2}\alpha)^{2})$ to be infinitely divisible for all $\alpha\in R^{1}$ is that \[ \Ga_{i,i}\geq \frac{c_{i}}{c_{j}}\Ga_{i,j}>0\qquad\forall 1\le i\ne j\le 2.\] In this paper we show that when this does not hold there exists an $0<\alpha_{0}<\ff $ such that $G=((G_{1}+c_{1}\alpha)^{2},(G_{2}+c_{2}\alpha)^{2})$ is infinitely divisible for all $|\alpha|\leq \alpha_{0}$ but not for any $|\al|> \al_{0}$. http://arxiv.org/abs/0806.3188 --------------------------------------------------------------- 7201. RENEWAL SERIES AND SQUARE-ROOT BOUNDARIES FOR BESSEL PROCESSES Nathanael Enriquez (MODAL'X) and Christophe Sabot (ICJ) and Marc Yor (PMA and IUF) We show how a description of Brownian exponential functionals as a renewal series gives access to the law of the hitting time of a square-root boundary by a Bessel process. This extends classical results by Breiman and Shepp, concerning Brownian motion, and recovers by different means, extensions for Bessel processes, obtained independently by Delong and Yor. http://arxiv.org/abs/0806.3197 --------------------------------------------------------------- 7202. HYDRODYNAMIC LIMIT OF GRADIENT EXCLUSION PROCESSES WITH CONDUCTANCES Tertuliano Franco and Claudio Landim Fix a strictly increasing right continuous with left limits function $W: \bb R \to \bb R$ and a smooth function $\Phi : [l,r] \to \bb R$, defined on some interval $[l,r]$ of $\bb R$, such that $0