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 $1/2