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 Deni