From pas at www.economia.unimi.it Thu Dec 30 12:03:38 2004 From: pas at www.economia.unimi.it (pas@www.economia.unimi.it) Date: Thu Dec 30 17:49:35 2004 Subject: [Pas] Probability Abstract 84 Message-ID: <755CABCD-5A52-11D9-93EA-000A95C87F66@unimi.it> December 30, 2004 Letter 84 Dear Colleagues, I am happy to distribute my first Probability Abstract Service Letter. This letter actually contains abstracts from the ArXiv archive only but there are plans to fill the PAS archive from different sources (like departmental working paper series) in the near future as well as re-enable abstract submissions directly to the PAS archive again (even if submitting to ArXiv directly is always preferable). The new PAS Web site, http://www.economia.unimi.it/PAS/ already exists and soon few new services will be announced. Thanks to Chris Burdzy for the amount of work he has done till now. I whish you all a happy new year with a thought to the victims of south-east asian earthquake. Stefano M. Iacus %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --------------------------------------------------------------- 2957. STEIN'S METHOD AND MINIMUM PARSIMONY DISTANCE AFTER SHUFFLES Jason Fulman Motivated by Bourque and Pevzner's simulation study of the parsimony method for studying genome rearrangement, Berestycki and Durrett used techniques from random graph theory to prove that the minimum parsimony distance after iterating the random transposition shuffle undergoes a transition from Poisson to normal behavior. This paper establishes an analogous result for minimum parsimony distance after iterates of riffle shuffles or iterates of riffle shuffles and cuts. The analysis is elegant and uses different tools: Stein's method and generating functions. A useful technique which emerges is that of making a problem more tractable by adding extra symmetry, then using Stein's method to exploit the symmetry in the modified problem, and from this deducing information about the original problem. http://front.math.ucdavis.edu/math.PR/0410622 --------------------------------------------------------------- 2958. CONVERGENCE OF MARKOV PROCESSES NEAR SADDLE FIXED POINTS Amanda G. Turner We consider sequences of Markov processes in two dimensions whose fluid limit is a stable solution of an ordinary differential equation of the form dx/dt = b(x), where the linear part of b(x) has eigenvalues -mu and lambda for some lambda, mu > 0. Here the processes are indexed so that the variance of the fluctuations is inversely proportional to N. The simplest example arises from the OK Corral gunfight model which was formulated by Williams and McIlroy (1998) and studied by Kingman (1999). These processes exhibit their most interesting behaviour at times of order log N so it is necessary to establish a fluid limit that is valid for large times. We find that this limit is inherently random and obtain its distribution. Using this, it is possible to derive scaling limits for the points where these processes hit straight lines through the origin, and the minimal distance from the origin that they can attain. The power of N that gives the appropriate scaling somewhat surprisingly turns out to be mu / 2(lamba + mu). http://front.math.ucdavis.edu/math.PR/0412051 --------------------------------------------------------------- 2959. AN UMBRAL SETTING FOR CUMULANTS AND FACTORIAL MOMENTS E. Di Nardo and D. Senato We provide an algebraic setting for cumulants and factorial moments through the classical umbral calculus. Main tools are the compositional inverse of the unity umbra, connected with the logarithmic power series, and a new umbra here introduced, the singleton umbra. Various formulae are given expressing cumulants, factorial moments and central moments by umbral functions. http://front.math.ucdavis.edu/math.PR/0412052 --------------------------------------------------------------- 2960. UMBRAL NATURE OF THE POISSON RANDOM VARIABLES E. Di Nardo and D. Senato Extending the rigorous presentation of the classical umbral calculus given by Rota and Taylor in 1994, the so-called partition polynomials are interpreted with the aim to point out the umbral nature of the Poisson random variables. Among the new umbrae introduced, the main tool is the partition umbra that leads also to a simple expression of the functional composition of the exponential power series. Moreover a new short proof of the Lagrange inversion formula is given. http://front.math.ucdavis.edu/math.PR/0412054 --------------------------------------------------------------- 2961. ON RADIAL STOCHASTIC LOEWNER EVOLUTION IN MULTIPLY CONNECTED DOMAINS Robert O. Bauer and Roland M. Friedrich We discuss the extension of radial SLE to multiply connected planar domains. First, we extend Loewner's theory of slit mappings to multiply connected domains by establishing the radial Komatu-Loewner equation, and show that a simple curve from the boundary to the bulk is encoded by a motion on moduli space and a motion on the boundary of the domain. Then, we show that the vector-field describing the motion of the moduli is Lipschitz. We explain why this implies that "consistent," conformally invariant random simple curves are described by multidimensional diffusions, where one component is a motion on the boundary, and the other component is a motion on moduli space. We argue what the exact form of this diffusion is (up to a single real parameter $\kappa$) in order to model boundaries of percolation clusters. Finally, we show that this moduli diffusion leads to random non-self-crossing curves satisfying the locality property if and only if $\kappa=6$. http://front.math.ucdavis.edu/math.PR/0412060 --------------------------------------------------------------- 2962. GENERALIZED CAUCHY IDENTITIES, TREES AND MULTIDIMENSIONAL BROWNIAN MOTIONS. PART I: BIJECTIVE PROOF OF GENERALIZED CAUCHY IDENTITIES Piotr Sniady In this series of articles we study connections between combinatorics of multidimensional generalizations of Cauchy identity and continuous objects such as multidimensional Brownian motions and Brownian bridges. In Part I of the series we present a bijective proof of multidimensional generalizations of the Cauchy identity. Our bijection uses oriented planar trees equipped with some linear orders. http://front.math.ucdavis.edu/math.CO/0412043 --------------------------------------------------------------- 2963. THE CHROMATIC NUMBER OF RANDOM REGULAR GRAPHS Dimitris Achlioptas and Cristopher Moore Given any integer d >= 3, let k be the smallest integer such that d < 2k log k. We prove that with high probability the chromatic number of a random d-regular graph is k, k+1, or k+2, and that if (2k-1) \log k < d < 2k \log k then the chromatic number is either k+1 or k+2. http://front.math.ucdavis.edu/cond-mat/0407278 --------------------------------------------------------------- 2964. NONNEGATIVE MATRIX FACTORIZATION AND I-DIVERGENCE ALTERNATING MINIMIZATION Lorenzo Finesso and Peter Spreij In this paper we consider the Nonnegative Matrix Factorization (NMF) problem: given an (elementwise) nonnegative matrix $V \in \R_+^{m\times n}$ find, for assigned $k$, nonnegative matrices $W\in\R_+^{m\times k}$ and $H\in\R_+^{k\times n}$ such that $V=WH$. Exact, non trivial, nonnegative factorizations do not always exist, hence it is interesting to pose the approximate NMF problem. The criterion which is commonly employed is I-divergence between nonnegative matrices. The problem becomes that of finding, for assigned $k$, the factorization $WH$ closest to $V$ in I-divergence. An iterative algorithm, EM like, for the construction of the best pair $(W, H)$ has been proposed in the literature. In this paper we interpret the algorithm as an alternating minimization procedure \`a la Csisz\'ar-Tusn\'ady and investigate some of its stability properties. NMF is widespreading as a data analysis method in applications for which the positivity constraint is relevant. There are other data analysis methods which impose some form of nonnegativity: we discuss here the connections between NMF and Archetypal Analysis. http://front.math.ucdavis.edu/math.OC/0412070 --------------------------------------------------------------- 2965. RANDOM PARTITIONS APPROXIMATING THE COALESCENCE OF LINEAGES DURING A SELECTIVE SWEEP Jason Schweinsberg and Rick Durrett When a beneficial mutation occurs in a population, the new, favored allele may spread to the entire population. This process is known as a selective sweep. Suppose we sample $n$ individuals at the end of a selective sweep. If we focus on a site on the chromosome that is close to the location of the beneficial mutation, then many of the lineages will likely be descended from the individual that had the beneficial mutation, while others will be descended from a different individual because of recombination between the two sites. We introduce two approximations for the effect of a selective sweep. The first one is simple but not very accurate: flip $n$ independent coins with probability $p$ of heads and say that the lineages whose coins come up heads are those that are descended from the individual with the beneficial mutation. A second approximation, which is related to Kingman's paintbox construction, replaces the coin flips by integer-valued random variables and leads to very accurate results. http://front.math.ucdavis.edu/math.PR/0411069 --------------------------------------------------------------- 2966. ON SKOROHOD SPACES AS UNIVERSAL SAMPLE PATH SPACES Oliver Delzeith The paper presents a factorization theorem for a certain class of stochastic processes. Skorohod spaces carry the rich structure of standard Borel spaces and appear to be suitable universal sample path spaces. We show that, if $\xi$ is a RCLL stochastic process with values in a complete separable metric space $E$, any other RCLL stochastic process $X$ adapted to the filtration induced by $\xi$ factors through the Skorohod space $D_E[0,\infty)$. This can be understood as an extension of a stochastic process to a standard Borel space enjoying nice properties. Moreover, the trajectories of the factorized stochastic process defined on $D_E[0,\infty)$ inherit the properties of being continuous, non-decreasing, and of bounded variation, resp., from those of $X$. Considering situations which are invariant under the factorization procedure, the main theorem is a reduction tool to assume the underlying measurable space be a standard Borel space. In an example, we pick the existence theorem of regular conditional probabilities on standard Borel spaces to simplify a conditional expectation appearing in stochastic control problems. http://front.math.ucdavis.edu/math.PR/0412092 --------------------------------------------------------------- 2967. A MASS TRANSFERENCE PRINCIPLE AND THE DUFFIN-SCHAEFFER CONJECTURE FOR HAUSDORFF MEASURES Victor Beresnevich and Sanju Velani A Hausdorff measure version of the Duffin-Schaeffer conjecture in metric number theory is introduced and discussed. The general conjecture is established modulo the original conjecture. The key result is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for $\limsup$ subsets of $\R^k$ to Hausdorff measure theoretic statements. In view of this, the Lebesgue theory of $\limsup $ sets is shown to underpin the general Hausdorff theory. This is rather surprising since the latter theory is viewed to be a subtle refinement of the former. http://front.math.ucdavis.edu/math.NT/0412141 --------------------------------------------------------------- 2968. DESTRUCTION OF VERY SIMPLE TREES James Allen Fill and Nevin Kapur and Alois Panholzer We consider the total cost of cutting down a random rooted tree chosen from a family of so-called very simple trees (which include ordered trees, $d$-ary trees, and Cayley trees); these form a subfamily of simply generated trees. At each stage of the process an edge is chose at random from the tree and cut, separating the tree into two components. In the one-sided variant of the process the component not containing the root is discarded, whereas in the two-sided variant both components are kept. The process ends when no edges remain for cutting. The cost of cutting an edge from a tree of size $n$ is assumed to be $n^\alpha$. Using singularity analysis and the method of moments, we derive the limiting distribution of the total cost accrued in both variants of this process. A salient feature of the limiting distributions obtained (after normalizing in a family-specific manner) is that they only depend on $\alpha$. http://front.math.ucdavis.edu/math.PR/0412155 --------------------------------------------------------------- 2969. RANDOM PARTITIONING PROBLEMS INVOLVING POISSON POINT PROCESSES ON THE INTERVAL Thierry Huillet (LPTM) Suppose some random resource (energy, mass or space) $\chi \geq 0$ is to be shared at random between (possibly infinitely many) species (atoms or fragments). Assume ${\Bbb E}\chi =\theta <\infty $ and suppose the amount of the individual share is necessarily bounded from above by 1. This random partitioning model can naturally be identified with the study of infinitely divisible random variables with L\'{e}vy measure concentrated on the interval% $.$ Special emphasis is put on these special partitioning models in the Poisson-Kingman class. The masses attached to the atoms of such partitions are sorted in decreasing order. Considering nearest- neighbors spacings yields a partition of unity which also deserves special interest. For such partition models, various statistical questions are addressed among which: correlation structure, cumulative energy of the first $K$ largest items, partition function, threshold and covering statistics, weighted partition, R\'{e}nyi's, typical and size-biased fragments size. Several physical images are supplied. When the unbounded L\'{e}vy measure of $\chi $ is $\theta x^{-1}\cdot {\bf I}% (x\in (0,1)) dx$, the spacings partition has Griffiths-Engen-McCloskey or GEM$(\theta) $ distribution and $% \chi $ follows Dickman distribution. The induced partition models have many remarkable peculiarities which are outlined. The case with finitely many (Poisson) fragments in the partition law is also briefly addressed. Here, the L\'{e}vy measure is bounded. http://front.math.ucdavis.edu/cond-mat/0412166 --------------------------------------------------------------- 2970. DEVROYE INEQUALITY FOR A CLASS OF NON-UNIFORMLY HYPERBOLIC DYNAMICAL SYSTEMS J.-R. Chazottes and P. Collet and B. Schmitt In this paper, we prove an inequality, which we call "Devroye inequality", for a large class of non-uniformly hyperbolic dynamical systems (M,f). This class, introduced by L.-S. Young, includes families of piece-wise hyperbolic maps (Lozi-like maps), scattering billiards (e.g., planar Lorentz gas), unimodal and H{\'e}non-like maps. Devroye inequality provides an upper bound for the variance of observables of the form K(x,f(x),...,f^{n-1}(x)), where K is any separately Holder continuous function of n variables. In particular, we can deal with observables which are not Birkhoff averages. We will show in \cite{CCS} some applications of Devroye inequality to statistical properties of this class of dynamical systems. http://front.math.ucdavis.edu/math.DS/0412166 --------------------------------------------------------------- 2971. STATISTICAL CONSEQUENCES OF DEVROYE INEQUALITY FOR PROCESSES. APPLICATIONS TO A CLASS OF NON-UNIFORMLY HYPERBOLIC DYNAMICAL SYSTEMS J.-R. Chazottes and P. Collet and B. Schmitt In this paper, we apply Devroye inequality to study various statistical estimators and fluctuations of observables for processes. Most of these observables are suggested by dynamical systems. These applications concern the co-variance function, the integrated periodogram, the correlation dimension, the kernel density estimator, the speed of convergence of empirical measure, the shadowing property and the almost-sure central limit theorem. We proved in \cite{CCS} that Devroye inequality holds for a class of non-uniformly hyperbolic dynamical systems introduced in \cite{young}. In the second appendix we prove that, if the decay of correlations holds with a common rate for all pairs of functions, then it holds uniformly in the function spaces. In the last appendix we prove that for the subclass of one-dimensional systems studied in \cite{young} the density of the absolutely continuous invariant measure belongs to a Besov space. http://front.math.ucdavis.edu/math.DS/0412167 --------------------------------------------------------------- 2972. THE FOREGOUND-BACKGROUND PROCESSOR SHARING QUEUE: AN OVERVIEW Misja Nuyens We give an overview of the results in the literature on single-server queues with the FB discipline. The FB discipline gives service to the customer that has received the least amount of service. This not so well-known discipline has some appealing features, and performs well for heavy-tailed service times. We describe results on the queue length, sojourn time, and the influence of variability in the service times. http://front.math.ucdavis.edu/math.PR/0412182 --------------------------------------------------------------- 2973. A PROBABILISTIC ANALYSIS OF SOME TREE ALGORITHMS Hanene Mohamed (RAP UR-R) and Philippe Robert (RAP UR-R) In this paper a general class of tree algorithms is analyzed. It is shown that, by using an appropriate probabilistic representation of the quantities of interest, the asymptotic behavior of these algorithms can be obtained quite easily without resorting to complex analysis techniques as it is usually the case. This approach gives a unified probabilistic treatment of these questions. It simplifies and extends some of the results known in this domain. http://front.math.ucdavis.edu/math.PR/0412188 --------------------------------------------------------------- 2974. A CONTINUOUS STOCHASTIC MATURATION MODEL Djalil Chafai (LSProba and Upte Umr Inra/Envt 181) and Didier Concordet (LSProba, Upte Umr Inra/Envt 181) We present a continuous time model of maturation and survival, obtained as the limit of a compartmental evolution model when the number of compartments tends to infinity. We establish in particular an explicit formula for the law of the system output under inhomogeneous killing and when the input follows a time-inhomogeneous Poisson process. Identifiability issues are discussed, and an application to the modelling of the toxicity of anti-cancer drugs is given. Such models can be seen in particular as generalisations of previous works of Jacquez & Simon and Schuhmacher & Thieme. http://front.math.ucdavis.edu/math.PR/0412193 --------------------------------------------------------------- 2975. ON LOCAL MARTINGALE AND ITS SUPREMUM: HARMONIC FUNCTIONS AND BEYOND Jan Obloj (PMA and Mimuw) and Marc Yor (PMA) We discuss certain facts involving a continuous local martingale $N$ and its supremum $\bar{N}$. A complete characterization of $(N,\bar{N})$-harmonic functions is proposed. This yields an important family of martingales, the usefulness of which is demonstrated, by means of examples involving the Skorokhod embedding problem, bounds on the law of the supremum, or the local time at 0, of a martingale with a fixed terminal distribution, or yet in some Brownian penalization problems. In particular we obtain new bounds on the law of the local time at 0, which involve the excess wealth order. http://front.math.ucdavis.edu/math.PR/0412196 --------------------------------------------------------------- 2976. A COALESCENT MODEL FOR THE EFFECT OF ADVANTAGEOUS MUTATIONS ON THE GENEALOGY OF A POPULATION Rick Durrett and Jason Schweinsberg When an advantageous mutation occurs in a population, the favorable allele may spread to the entire population in a short time, an event known as a selective sweep. As a result, when we sample $n$ individuals from a population and trace their ancestral lines backwards in time, many lineages may coalesce almost instantaneously at the time of a selective sweep. We show that as the population size goes to infinity, this process converges to a coalescent process called a coalescent with multiple collisions. A better approximation for finite populations can be obtained using a coalescent with simultaneous multiple collisions. We also show how these coalescent approximations can be used to get insight into how beneficial mutations affect the behavior of statistics that have been used to detect departures from the usual Kingman's coalescent. http://front.math.ucdavis.edu/math.PR/0411071 --------------------------------------------------------------- 2977. ROUGHENING AND INCLINATION OF COMPETITION INTERFACES Pablo A. Ferrari and James B. Martin and Leandro P. R. Pimentel The competition interface between two growing ``Young clusters'' (diagrams), in a two-dimensional random cone, is mapped to the path of a second-class particle in the one-dimensional totally asymmetric simple exclusion process. Using the asymptotics of the second class particle and hydrodynamic limits for the exclusion process (Burgers equation), we show that the behavior of the competition interface depends on the angle of the cone: for angles in [180^o, 270^o) the competition interface has a deterministic inclination, while for angles in [90^o,180^o) the inclination is random. We relate the competition model to a model of random directed polymers, and obtain some partial results for the fluctuations of the competition interface. http://front.math.ucdavis.edu/math.PR/0412198 --------------------------------------------------------------- 2978. LARGE DEVIATIONS FOR ROUGH PATHS OF THE FRACTIONAL BROWNIAN MOTION Annie Millet and Marta Sanz-Sol\'e Starting from the construction of a geometric rough path associated with a fractional Brownian motion with Hurst parameter $H\in]{1/4}, {1/2}[$ given by Coutin and Qian (2002), we prove a large deviation principle in the space of geometric rough paths, extending classical results on Gaussian processes. As a by-product, geometric rough paths associated to elements of the reproducing kernel Hilbert space of the fractional Brownian motion are obtained and an explicit integral representation is given. http://front.math.ucdavis.edu/math.PR/0412200 --------------------------------------------------------------- 2979. ANALYTICAL PROBABILISTIC APPROACH TO THE GROUND STATE OF LATTICE QUANTUM Massimo Ostilli and Carlo Presilla We present a large deviation analysis of a recently proposed probabilistic approach to the study of the ground-state properties of lattice quantum systems. The ground-state energy, as well as the correlation functions in the ground state, are exactly determined as a series expansion in the cumulants of the multiplicities of the potential and hopping energies assumed by the system during its long-time evolution. Once these cumulants are known, even at a finite order, our approach provides the ground state analytically as a function of the Hamiltonian parameters. A scenario of possible applications of this analyticity property is discussed. http://front.math.ucdavis.edu/cond-mat/0412157 --------------------------------------------------------------- 2980. THE COMPACT SUPPORT PROPERTY FOR MEASURE-VALUED DIFFUSIONS Ross G. Pinsky The purpose of this article is to give a rather thorough understanding of the compact support property for measure-valued diffusion processes corresponding to semi-linear equations of the form \[& u_t=Lu+\beta u-\alpha u^p \text{in} R^d\times (0,\infty), p\in(1,2]; &u(x,0)=f(x) \text{in} R^d; &u(x,t)\ge0 \text{in} R^d\times[0,\infty). \] In particular, we shall investigate how the interplay between the underlying motion (the diffusion process corresponding to $L$) and the branching affects the compact support property. In \cite{EP99}, the compact support property was shown to be equivalent to a certain analytic criterion concerning uniqueness of the Cauchy problem for the semilinear parabolic equation related to the measured valued diffusion. In a subsequent paper \cite{EP03}, this analytic property was investigated purely from the point of view of partial differential equations. Some of the results obtained in this latter paper yield interesting results concerning the compact support property. In this paper, the results from \cite{EP03} that are relevant to the compact support property are presented, sometimes with extensions. These results are interwoven with new results and some informal heuristics. Taken together, they yield a fairly comprehensive picture of the compact support property. \it Inter alia\rm, we show that the concept of a measure-valued diffusion \it hitting\rm a point can be investigated via the compact support property, and suggest an alternate proof of a result concerning the hitting of points by super-Brownian motion. http://front.math.ucdavis.edu/math.PR/0412246 --------------------------------------------------------------- 2981. EXPLICIT CHARACTERIZATION OF THE SUPER-REPLICATION STRATEGY IN FINANCIAL MARKETS WITH PARTIAL TRANSACTION COSTS Imen Bentahar (CEREMADE) and Bruno Bouchard (CREST and Lfa and Pma) We consider a multivariate financial market with transaction costs and study the problem of finding the minimal initial capital needed to hedge, without risk, European-type contingent claims. The model is similar to the one considered in Bouchard and Touzi (2000), except that some of the assets can be exchanged freely, i.e. without paying transaction costs. In this context, we generalize the result of the above paper and prove that the value of this stochastic control problem is given by the cost of the cheapest hedging strategy in which the number of non-freely exchangeable assets is kept constant over time. http://front.math.ucdavis.edu/math.PR/0412247 --------------------------------------------------------------- 2982. ON AUTOMORPHISMS OF TYPE II ARVESON SYSTEMS (PROBABILISTIC APPROACH) Boris Tsirelson A counterexample to the conjecture that the automorphisms of an arbitrary Arveson system act transitively on its normalized units. http://front.math.ucdavis.edu/math.OA/0411062 --------------------------------------------------------------- 2983. MINIMAL SPANNING FORESTS Russell Lyons and Yuval Peres and and Oded Schramm We study minimal spanning forests in infinite graphs, which are weak limits of minimal spanning trees from finite subgraphs corresponding to i.i.d. random labels on the edges. These limits can be taken with free or wired boundary conditions, and are denoted $\fmsf$ (free minimal spanning forest) and $\wmsf$ (wired minimal spanning forest), respectively. The $\wmsf$ is the union of the trees that arise from invasion percolation started at all vertices. We show that on any Cayley graph where critical percolation has no infinite clusters, all the component trees in the $\wmsf$ have one end a.s. In $\Z^d$ this was proved by \ref b.Alexander:MSF/, but a different method is needed for the nonamenable case. We show that on any connected graph, the union of the $\fmsf$ and independent Bernoulli percolation (with arbitrarily small parameter) is a.s. connected. In conjunction with a recent result of Gaboriau, this implies that in any Cayley graph, the expected degree of the $\fmsf$ is at least the expected degree of the $\fsf$ (the weak limit of uniform spanning trees). We show that on any graph, each component tree in the $\wmsf$ has $\pc = 1$ a.s., where $\pc$ denotes the critical probability for having an infinite cluster in Bernoulli percolation. We show that the number of infinite clusters for Bernoulli($\pu$) percolation is at most the number of components of the $\fmsf$, where $\pu$ denotes the critical probability for having a unique infinite cluster. http://front.math.ucdavis.edu/math.PR/0412263 --------------------------------------------------------------- 2984. EMPIRICAL PROCESSES OF DEPENDENT RANDOM VARIABLES Wei Biao Wu Empirical processes for stationary, causal sequences are considered. We establish empirical central limit theorems for classes of indicators of left half lines, absolutely continuous functions and piecewise differentiable functions. Sample path properties of empirical distribution functions are also discussed. The results are applied to linear processes and Markov chains. http://front.math.ucdavis.edu/math.ST/0412267 --------------------------------------------------------------- 2985. M-ESTIMATION OF LINEAR MODELS WITH DEPENDENT ERRORS Wei Biao Wu We study the asymptotic behavior of M-estimates of regression parameters in multiple linear models where errors are dependent random variables. A Bahadur representation of the M-estimates is derived and a central limit theorem is established. The results are applied to linear models with errors being short-range dependent linear processes, heavy-tailed linear processes and some widely used nonlinear time series. http://front.math.ucdavis.edu/math.ST/0412268 --------------------------------------------------------------- 2986. STATIONARY TRANSFORMATION OF INTEGRATED BROWNIAN MOTION Eugene Wong Consider an n-fold integrated Brownian motion. We show that a simple change in time and scale transforms it into a stationary Gaussian process. The collection of stationary processes so constructed not only constitutes an interesting family of processes, but their spectral representation is also useful in dealing with integrated Brownian motion. We illustrate this by deriving an explicit representation for the joint density function for a family of integrated Brownian motions and showing some of its properties. http://front.math.ucdavis.edu/math.PR/0412291 --------------------------------------------------------------- 2987. NARROW ESCAPE, PART I A. Singer and Z. Schuss and D. Holcman and R.S. Eisenberg A Brownian particle with diffusion coefficient $D$ is confined to a bounded domain of volume $V$ in $\rR^3$ by a reflecting boundary, except for a small absorbing window. The mean time to absorption diverges as the window shrinks, thus rendering the calculation of the mean escape time a singular perturbation problem. We construct an asymptotic approximation for the case of an elliptical window of large semi axis $a\ll V^{1/3}$ and show that the mean escape time is $E\tau\sim\ds{\frac{V}{2\pi Da}} K(e)$, where $e$ is the eccentricity of the ellipse; and $K(\cdot)$ is the complete elliptic integral of the first kind. In the special case of a circular hole the result reduces to Lord Rayleigh's formula $E\tau\sim\ds{\frac{V}{4aD}}$, which was derived by heuristic considerations. For the special case of a spherical domain, we obtain the asymptotic expansion $E\tau=\ds{\frac{V}{4aD}} [1+\frac{a}{R} \log \frac{R}{a} + O(\frac{a}{R}) ]$. This problem is important in understanding the flow of ions in and out of narrow valves that control a wide range of biological and technological function. http://front.math.ucdavis.edu/math-ph/0412048 --------------------------------------------------------------- 2988. NARROW ESCAPE, PART II: THE CIRCULAR DISK A. Singer and Z. Schuss and D. Holcman We consider Brownian motion in a circular disk $\Omega$, whose boundary $\p\Omega$ is reflecting, except for a small arc, $\p\Omega_a$, which is absorbing. As $\epsilon=|\partial \Omega_a|/|\partial \Omega|$ decreases to zero the mean time to absorption in $\p\Omega_a$, denoted $E\tau$, becomes infinite. The narrow escape problem is to find an asymptotic expansion of $E\tau$ for $\epsilon\ll1$. We find the first two terms in the expansion and an estimate of the error. The results are extended in a straightforward manner to planar domains and two-dimensional Riemannian manifolds that can be mapped conformally onto the disk. Our results improve the previously derived expansion for a general smooth domain, $E\tau = \ds{\frac{|\Omega|}{D\pi}}[\log\ds{\frac{1}{\epsilon}}+O(1)],$ ($D$ is the diffusion coefficient) in the case of a circular disk. We find that the mean first passage time from the center of the disk is $E[\tau | \x(0)=\mb{0}]=\ds{\frac{R^2}{D}}[\log\ds{\frac{1}{\epsilon}} + \log 2 +\ds{{1/4}} + O(\epsilon)]$. The second term in the expansion is needed in real life applications, such as trafficking of receptors on neuronal spines, because $\log\ds{\frac{1}{\epsilon}}$ is not necessarily large, even when $\epsilon$ is small. We also find the singular behavior of the probability flux profile into $\p\Omega_a$ at the endpoints of $\p\Omega_a$, and find the value of the flux near the center of the window. http://front.math.ucdavis.edu/math-ph/0412050 --------------------------------------------------------------- 2989. NARROW ESCAPE, PART III: RIEMANN SURFACES AND NON-SMOOTH DOMAINS A. Singer and Z. Schuss and D. Holcman We consider Brownian motion in a bounded domain $\Omega$ on a two-dimensional Riemannian manifold $(\Sigma,g)$. We assume that the boundary $\p\Omega$ is smooth and reflects the trajectories, except for a small absorbing arc $\p\Omega_a\subset\p\Omega$. As $\p\Omega_a$ is shrunk to zero the expected time to absorption in $\p\Omega_a$ becomes infinite. The narrow escape problem consists in constructing an asymptotic expansion of the expected lifetime, denoted $E\tau$, as $\epsilon=|\partial \Omega_a|_g/|\partial \Omega|_g\to0$. We derive a leading order asymptotic approximation $E\tau = \ds{\frac{|\Omega|_g}{D\pi}}[\log\ds{\frac{1}{\epsilon}}+O(1)]$. The order 1 term can be evaluated for simply connected domains on a sphere by projecting stereographically on the complex plane and mapping conformally on a circular disk. It can also be evaluated for domains that can be mapped conformally onto an annulus. This term is needed in real life applications, such as trafficking of receptors on neuronal spines, because $\log\ds{\frac{1}{\epsilon}}$ is not necessarily large, even when $\epsilon$ is small. If the absorbing window is located at a corner of angle $\alpha$, then $E\tau = \ds{\frac{|\Omega|_g}{D\alpha}}[\log\ds{\frac{1}{\epsilon}}+O(1)],$ if near a cusp, then $E\tau$ grows algebraically, rather than logarithmically. Thus, in the domain bounded between two tangent circles, the expected lifetime is $E\tau = \ds{\frac{|\Omega|}{(d^{-1}-1)D}}(\frac{1}{\epsilon} + O(1))$. http://front.math.ucdavis.edu/math-ph/0412051 --------------------------------------------------------------- 2990. OPTIMAL MASS TRANSPORTATION AND MATHER THEORY Patrick Bernard (IF) and Boris Buffoni (EPFL) We study optimal transportation of measures on compact manifolds for costs defined from convex Lagrangians. We prove that optimal transportation can be interpolated by measured Lipschitz laminations, or geometric currents. The methods are inspired from Mather theory on Lagrangian systems. We make use of viscosity solutions of the associated Hamilton-Jacobi equation in the spirit of Fathi's approach to Mather theory. http://front.math.ucdavis.edu/math.DS/0412299 --------------------------------------------------------------- 2991. SOME CONNECTIONS BETWEEN (SUB)CRITICAL BRANCHING MECHANISMS AND BERNSTEIN FUNCTIONS Jean Bertoin (PMA) and Bernard Roynette (IEC) and Marc Yor (PMA) We describe some connections, via composition, between two functional spaces: the space of (sub)critical branching mechanisms and the space of Bernstein functions. The functions ${\bf e}_\alpha: x\to x^{\alpha}$ where $x\geq0$ and $0<\alpha\leq 1/2$, and in particular the critical parameter $\alpha=1/2$, play a distinguished role. http://front.math.ucdavis.edu/math.PR/0412322 --------------------------------------------------------------- 2992. AN URN MODEL OF DIACONIS David Siegmund and Benjamin Yakir An urn model of Diaconis and some generalizations are discussed. A convergence theorem is proved that implies for Diaconis' model that the empirical distribution of balls in the urn converges with probability one to the uniform distribution. http://front.math.ucdavis.edu/math.PR/0412333 --------------------------------------------------------------- 2993. UNIFORM LARGE DEVIATIONS FOR THE NONLINEAR SCHRODINGER EQUATION WITH MULTIPLICATIVE NOISE Eric Gautier (IRMAR and Crest-Insee Laboratoire De Statistique) Uniform large deviations for the laws of the paths of the solutions of the stochastic nonlinear Schrodinger equation when the noise converges to zero are presented. The noise is a real multiplicative Gaussian noise. It is white in time and colored in space. The path space considered allows blow-up and is endowed with a topology analogue to a projective limit topology. Thus a large variety of large deviation principle may be deduced by contraction. As a consequence, asymptotics of the tails of the law of the blow-up time when the noise converges to zero are obtained. http://front.math.ucdavis.edu/math.AP/0412319 --------------------------------------------------------------- 2994. ON FINITE RANGE STABLE TYPE CONCENTRATION J.C. Breton and C. Houdr\'e The purpose of these notes is to further complete our understanding of the stable concentration phenomenon, by obtaining the finite range behavior of $P(F-E[F]\geq x)$, with $F=f(X)$ where $f$ is a Lipschitz function and $X$ is a stable random vector or with $F$ a stochastic functional on the Poisson space equipped with a stable L\'evy measure. http://front.math.ucdavis.edu/math.PR/0412334 --------------------------------------------------------------- 2995. CONSTRUCTION OF THE THERMODYNAMIC LIMIT MEASURE FOR THE PARKING PROCESS AND OTHER EXCLUSION SCHEMES ON $\MATHBB{Z}^{D}$ Thomas Logan Ritchie We provide an explicit construction for the thermodynamic limit measure for finite range exclusion schemes on $\mathbb{Z}^{d}$. By means thereof a strong law of large numbers for occupation densities is accomplished, and, amongst other results, the so called ``super-exponential'' (i.e. gamma) decay of pair-correlation functions is established. http://front.math.ucdavis.edu/math.PR/0412343 --------------------------------------------------------------- 2996. DIMENSION FREE AND INFINITE VARIANCE TAIL ESTIMATES ON POISSON SPACE J.C. Breton and C. Houdr\'e and N. Privault Concentration inequalities are obtained on Poisson space, for random functionals with finite or infinite variance. In particular, dimension free tail estimates and exponential integrability results are given for the Euclidean norm of vectors of independent functionals. In the finite variance case these results are applied to infinitely divisible random variables such as quadratic Wiener functionals, including L\'evy's stochastic area and the square norm of Brownian paths. In the infinite variance case, various tail estimates such as stable ones are also presented. http://front.math.ucdavis.edu/math.PR/0412346 --------------------------------------------------------------- 2997. IMPROVED LOWER BOUNDS FOR THE CRITICAL PROBABILITY OF ORIENTED-BOND PERCOLATION IN TWO DIMENSIONS Thomas Logan Ritchie and Vladimir Belitsky We present a coupled decreasing sequence of random walks on $ \mathbb Z $ that dominates the edge process of oriented-bond percolation in two dimensions. Using the concept of "random walk in a strip ", we construct an algorithm that generates an increasing sequence of lower bounds that converges to the critical probability of oriented-bond percolation. Numerical calculations of the first ten lower bounds thereby generated lead to an improved,i.e. higher, rigorous lower bound to this critical probability, viz. $p_{c} \geq 0.63328 $. Finally a computer simulation technique is presented; the use thereof establishes 0.64450 as a non-rigorous five-digit-precision (lower) estimate for $p_{c}$. http://front.math.ucdavis.edu/math.PR/0412348 --------------------------------------------------------------- 2998. Q-MARKOV RANDOM PROBABILITY MEASURES AND THEIR POSTERIOR DISTRIBUTIONS Raluca Balan In this paper, we use the Markov property introduced in Balan and Ivanoff (J. Theor. Probab. 15, 2002, 553-588) for set-indexed processes and we prove that a Markov prior distribution leads to a Markov posterior distribution. In particular, by proving that a neutral to the right prior distribution leads to a neutral to the right posterior distribution, we extend a fundamental result of Doksum (Ann. Probab. 2,1974, 183-201) to arbitrary sample spaces. http://front.math.ucdavis.edu/math.PR/0412349 --------------------------------------------------------------- 2999. A MARKOV PROPERTY FOR SET-INDEXED PROCESSES Raluca Balan and Gail Ivanoff We consider a type of Markov property for set-indexed processes which is satisfied by all processes with independent increments and which allows us to introduce a transition system theory leading to the construction of the process. A set-indexed generator is defined such that it completely characterizes the distribution of the process. http://front.math.ucdavis.edu/math.PR/0412350 --------------------------------------------------------------- 3000. STEIN ESTIMATION FOR INFINITELY DIVISIBLE LAWS R. Averkamp and C. Houdr\'e Unbiased risk estimation, \`a la Stein, is studied for infinitely divisible laws with finite second moment. http://front.math.ucdavis.edu/math.ST/0412345 --------------------------------------------------------------- 3001. NONPARAMETRIC ESTIMATION FOR LEVY PROCESSES WITH A VIEW TOWARDS MATHEMATICAL FINANCE Enrique Figueroa-Lopez and Christian Houdre Nonparametric methods for the estimation of the Levy density of a Levy process are developed. Estimators that can be written in terms of the ``jumps'' of the process are introduced, and so are discrete-data based approximations. A model selection approach made up of two steps is investigated. The first step consists in the selection of a good estimator from a linear model of proposed Levy densities, while the second is a data-driven selection of a linear model among a given collection of linear models. By providing lower bounds for the minimax risk of estimation over Besov Levy densities, our estimators are shown to achieve the ``best'' rate of convergence. A numerical study for the case of histogram estimators and for variance Gamma processes, models of key importance in risky asset price modeling driven by Levy processes, is presented. http://front.math.ucdavis.edu/math.ST/0412351 --------------------------------------------------------------- 3002. IMPROVING ON BOLD PLAY WHEN THE GAMBLER IS RESTRICTED Jason Schweinsberg Suppose a gambler starts with a fortune in (0,1) and wishes to attain a fortune of 1 by making a sequence of bets. Assume thay whenever the gambler stakes the amount s, the gambler's fortune increases by s with probability w and decreases by s with probability 1 - w, where w < 1/2. Dubins and Savage showed that the optimal strategy, which they called "bold play", is always to stake min{f, 1-f}, where f is the gambler's current fortune. Here we consider the problem in which the gambler may stake no more than l at one time. We show that the bold strategy of always betting min{l, f, 1-f} is not optimal if l is irrational, extending a result of Heath, Pruitt, and Sudderth. http://front.math.ucdavis.edu/math.PR/0412362 --------------------------------------------------------------- 3003. A GUE CENTRAL LIMIT THEOREM AND UNIVERSALITY OF DIRECTED FIRST AND LAST PASSAGE SITE PERCOLATION Jinho Baik and Toufic M. Suidan We prove a GUE central limit theorem for random variables with finite fourth moment. We apply this theorem to prove that the directed first and last passage percolation problems in thin rectangles exhibit universal fluctuations given by the Tracy-Widom law. In addition, we conjecture a precise value for the time constant in the general first and last passage problems. http://front.math.ucdavis.edu/math.PR/0412369 --------------------------------------------------------------- 3004. LONGEST COMMON SUBSEQUENCES AND THE BERNOULLI MATCHING MODEL: NUMERICAL WORK AND ANALYSES OF THE R-REACH SIMPLIFICATION Jonah Blasiak The expected length of longest common subsequences is a problem that has been in the literature for at least twenty five years. Determining the limiting constants \gamma_k appears to be quite difficult, and the current best bounds leave much room for improvement. Boutet de Monvel explores an independent version of the problem he calls the Bernoulli Matching model. He explores this problem and its relation to the longest common subsequence problem. This paper continues this pursuit by focusing on a simplification we term r-reach. For the string model, L_r(u,v) is the longest common subsequence of u and v given that each matched pair of letters is no more than r letters apart. http://front.math.ucdavis.edu/math.PR/0412375 --------------------------------------------------------------- 3005. NOISE STABILITY OF WEIGHTED MAJORITY Yuval Peres Benjamini, Kalai and Schramm (2001) showed that weighted majority functions of $n$ independent unbiased bits are uniformly stable under noise: when each bit is flipped with probability $\epsilon$, the probability $p_\epsilon$ that the weighted majority changes is at most $C\epsilon^{1/4}$. They asked what is the best possible exponent that could replace 1/4. We prove that the answer is 1/2. The upper bound obtained for $p_\epsilon$ is within a factor of $\sqrt{\pi/2}+o(1)$ from the known lower bound when $\epsilon \to 0$ and $n\epsilon\to \infty$. http://front.math.ucdavis.edu/math.PR/0412377 --------------------------------------------------------------- 3006. THE ESCAPE MODEL ON A HOMOGENEOUS TREE G. Kordzakhia There are two types of particles interacting on a homogeneous tree of degree d + 1. The particles of the first type colonize the empty space with exponential rate 1, but cannot take over the vertices that are occupied by the second type. The particles of the second type spread with exponential rate \lambda. They colonize the neighboring vertices that are either vacant or occupied by the representatives of the opposite type, and annihilate the particles of the type 1 as they reach them. There exists a critical value \lambda_c =(2d - 1) + \sqrt{(2d -1)^2 -1} such that the first type survives with positive probability for \lambda < \lambda_c, and dies out with probability one for \lambda > \lambda_c. We also find the growth profile which characterizes the rate of growth of the type 1 in the space-time on the event of survival. http://front.math.ucdavis.edu/math.PR/0412392 --------------------------------------------------------------- 3007. EXCITED RANDOM WALK IN ONE DIMENSION T. Antal (1 and 2) and S. Redner (2 and 1) ((1) Boston University and (2) CNLS and Los Alamos National Laboratory) We study the excited random walk, in which a walk that is at a site that contains cookies eats one cookie and then hops to the right with probability p and to the left with probability q=1-p. If the walk hops onto an empty site, there is no bias. For the 1-excited walk on the half-line (one cookie initially at each site), the probability of first returning to the starting point at time t scales as t^{-(2-p)}. Although the average return time to the origin is infinite for all p, the walk eats, on average, only a finite number of cookies until this first return when p<1/2. For the infinite line, the probability distribution for the 1-excited walk has an unusual anomaly at the origin. The positions of the leftmost and rightmost uneaten cookies can be accurately estimated by probabilistic arguments and their corresponding distributions have power-law singularities near the origin. The 2-excited walk on the infinite line exhibits peculiar features in the regime p>3/4, where the walk is transient, including a mean displacement that grows as t^{nu}, with nu>1/2 dependent on p, and a breakdown of scaling for the probability distribution of the walk. http://front.math.ucdavis.edu/math.PR/0412407 --------------------------------------------------------------- 3008. ON THE INVARIANT MEASURE OF A POSITIVE RECURRENT DIFFUSION IN R Michele L. Baldini Given an one-dimensional positive recurrent diffusion governed by the Stratonovich SDE \[ X_t=x+\int_0^t\sigma(X_s)\strat db(s)+\int_0^t m(X_s) ds, \] we show that the associated stochastic flow of diffeomorphisms focuses as fast as $ \mathrm{exp}(-2t\int_{R}\frac{m^2}{\sigma^2} d\Pi)$, where $d\Pi$ is the finite stationary measure. Moreover, if the drift is reversed and the diffeomorphism is inverted, then the path function so produced tends, independently of its starting point, to a single (random) point whose distribution is $d\Pi$. Applications to stationary solutions of $X_t$, asymptotic behavior of solutions of SPDEs and random attractors are offered. http://front.math.ucdavis.edu/math.PR/0412410 --------------------------------------------------------------- 3009. NULL FLOWS, POSITIVE FLOWS AND THE STRUCTURE OF STATIONARY SYMMETRIC STABLE PROCESSES Gennady Samorodnitsky This paper elucidates the connection between stationary symmetric alpha-stable processes with 0 tc, we let simultaneously lambda tend to 0 and m to infinity, the probability that some tree at distance smaller than m from O is burnt before time t goes to 1. However, we show that under a percolation-like assumption (which we can not prove but believe to be true) this intuition is false. We compare with the case where the square lattice is replaced by the directed binary tree, and pose some natural open problems. http://front.math.ucdavis.edu/math.PR/0412488 --------------------------------------------------------------- 3017. EXCHANGEABLE GIBBS PARTITIONS AND STIRLING TRIANGLES Alexander Gnedin and Jim Pitman For two collections of nonnegative and suitably normalised weights $\W=(\W_j)$ and $\V=(\V_{n,k})$, a probability distribution on the set of partitions of the set $\{1,...,n\}$ is defined by assigning to a generic partition $\{A_j, j\leq k\}$ the probability $\V_{n,k} \W_{|A_1|}... \W_{|A_k|}$, where $|A_j|$ is the number of elements of $A_j$. We impose constraints on the weights by assuming that the resulting random partitions $\Pi_n$ of $[n]$ are consistent as $n$ varies, meaning that they define an exchangeable partition of the set of all natural numbers. This implies that the weights $\W$ must be of a very special form depending on a single parameter $\alpha\in [-\infty,1]$. The case $\alpha=1$ is trivial, and for each value of $\alpha\neq 1$ the set of possible $\V$-weights is an infinite-dimensional simplex. We identify the extreme points of the simplex by solving the boundary problem for a generalised Stirling triangle. In particular, we show that the boundary is discrete for $-\infty\leq\alpha<0$ and continuous for $0\leq\alpha<1$. For $\alpha\leq 0$ the extremes correspond to the members of the Ewens-Pitman family of random partitions indexed by $(\alpha,\theta)$, while for $0<\alpha<1$ the extremes are obtained by conditioning an $(\alpha,\theta)$-partition on the asymptotics of the number of blocks of $\Pi_n$ as $n$ tends to infinity. http://front.math.ucdavis.edu/math.PR/0412494 --------------------------------------------------------------- 3018. FUNCTION-VALUED STOCHASTIC CONVOLUTIONS ARISING IN INTEGRODIFFERENTIAL EQUATIONS Anna Karczewska We study stochastic convolutions providing by fundamental solutions of a class of integrodifferential equations which interpolate the heat and the wave equations. We give sufficient condition for the existence of function--valued convolutions in terms of the covariance kernel of a noise given by spatially homogeneous Wiener process. http://front.math.ucdavis.edu/math.PR/0412495 --------------------------------------------------------------- 3019. MAXIMAL TYPE INEQUALITIES FOR LINEAR STOCHASTIC VOLTERRA EQUATIONS Anna Karczewska The paper is devoted to estimates for convolutions appearing in some class of stochastic Volterra equations. Two maximal inequalities and exponential tail estimate are provided. In the paper the fractional method of infinite dimensional stochastic calculus has been used. http://front.math.ucdavis.edu/math.PR/0412496 --------------------------------------------------------------- 3020. SHARP THRESHOLDS AND PERCOLATION IN THE PLANE Bela Bollobas and Oliver Riordan Recently, the authors showed that the critical probability for random Voronoi percolation in the plane is 1/2. A by-product of the method was a short proof of the Harris-Kesten Theorem concerning bond percolation in the planar square lattice. The aim of this paper is to show that the same techniques can be applied to many other planar percolation models, both to obtain short proofs of known results, and to prove new ones. http://front.math.ucdavis.edu/math.PR/0412510 --------------------------------------------------------------- 3021. STOCHASTIC VOLTERRA CONVOLUTION WITH L\'EVY PROCESS Anna Karczewska In the paper we study stochastic convolution appearing in Volterra equation driven by so called L\'evy process. By L\'evy process we mean a process with homogeneous independent increments, continuous in probability and cadlag. http://front.math.ucdavis.edu/math.PR/0411148 --------------------------------------------------------------- 3022. WHY DELANNOY NUMBERS? Cyril Banderier (LIPN) and Sylviane Schwer (LIPN) This article is not a research paper, but a little note on the history of combinatorics: We present here a tentative short biography of Henri Delannoy, and a survey of his most notable works. This answers to the question raised in the title, as these works are related to lattice paths enumeration, to the so-called Delannoy numbers, and were the first general way to solve Ballot-like problems. These numbers appear in probabilistic game theory, alignments of DNA sequences, tiling problems, temporal representation models, analysis of algorithms and combinatorial structures. http://front.math.ucdavis.edu/math.CO/0411128 --------------------------------------------------------------- 3023. GENERATING FUNCTIONS FOR KERNELS OF DIGRAPHS (ENUMERATION & ASYMPTOTICS FOR NIM GAMES) Cyril Banderier (LIPN) and Jean-Marie Le Bars (LIPN and GREYC) and Vlady Ravelomanana (LIPN) In this article, we study directed graphs (digraphs) with a coloring constraint due to Von Neumann and related to Nim-type games. This is equivalent to the notion of kernels of digraphs, which appears in numerous fields of research such as game theory, complexity theory, artificial intelligence (default logic, argumentation in multi-agent systems), 0-1 laws in monadic second order logic, combinatorics (perfect graphs)... Kernels of digraphs lead to numerous difficult questions (in the sense of NP-completeness, #P-completeness). However, we show here that it is possible to use a generating function approach to get new informations: we use technique of symbolic and analytic combinatorics (generating functions and their singularities) in order to get exact and asymptotic results, e.g. for the existence of a kernel in a circuit or in a unicircuit digraph. This is a first step toward a generatingfunctionology treatment of kernels, while using, e.g., an approach "a la Wright". Our method could be applied to more general "local coloring constraints" in decomposable combinatorial structures. http://front.math.ucdavis.edu/math.CO/0411138 --------------------------------------------------------------- 3024. INFINITE DIMENSIONAL ENTANGLED MARKOV CHAINS Francesco Fidaleo We continue the analysis of nontrivial examples of quantum Markov processes. This is done by applying the construction of entangled Markov chains obtained from classical Markov chains with infinite state--space. The formula giving the joint correlations arises from the corresponding classical formula by replacing the usual matrix multiplication by the Schur multiplication. In this way, we provide nontrivial examples of entangled Markov chains on $\bar{\cup_{J\subset Z} \bar{\otimes}_{J}F}^{C^{*}}$, $F$ being any infinite dimensional type $I$ factor, $J$ a finite interval of $Z$, and the bar the von Neumann tensor product between von Neumann algebras. We then have new nontrivial examples of quantum random walks which could play a r\^ole in quantum information theory. In view of applications to quantum statistical mechanics too, we see that the ergodic type of an entangled Markov chain is completely determined by the corresponding ergodic type of the underlying classical chain, provided that the latter admits an invariant probability distribution. This result parallels the corresponding one relative to the finite dimensional case. Finally, starting from random walks on discrete ICC groups, we exhibit examples of quantum Markov processes based on type $II_1$ von Neumann factors. http://front.math.ucdavis.edu/math.OA/0411202 --------------------------------------------------------------- 3025. ON THE SPECTRUM OF MARKOV SEMIGROUPS VIA SAMPLE PATH LARGE DEVIATIONS Irina Ignatiouk-Robert The essential spectral radius of a sub-Markovian process is defined as the infimum of the spectral radiuses of all local perturbations of the process. When the family of rescaled processes satisfies sample path large deviation principle, the spectral radius and the essential spectral radius are expressed in terms of the rate function. The paper is motivated by applications to reflected diffusions and jump Markov processes describing stochastic networks for which the sample path large deviation principle has been established and the rate function has been identified while essential spectral radius has not been calculated. http://front.math.ucdavis.edu/math.PR/0411221 --------------------------------------------------------------- 3026. DIRECTED POLYMERS IN RANDOM ENVIRONMENT ARE DIFFUSIVE AT WEAK DISORDER Francis Comets (PMA) and Nobuo Yoshida (DIVISION of Mathematics and Kyoto University) In this paper, we consider directed polymers in random environment with discrete space and time. For transverse dimension at least equal to 3, we prove that diffusivity holds for the path in the full weak disorder region, i.e., where the partition function differs from its annealed value only by a non-vanishing factor. Deep inside this region, we also show that the quenched averaged energy has fluctuations of order 1. In complete generality (arbitrary dimension and temperature), we prove monotonicity of the phase diagram in the temperature. http://front.math.ucdavis.edu/math.PR/0411223 --------------------------------------------------------------- 3027. POISSON CALCULUS FOR SPATIAL NEUTRAL TO THE RIGHT PROCESSES Lancelot F. James In this paper we consider classes of nonparametric priors on spaces of distribution functions and cumulative hazards that are based on extensions of the neutral to the right concept. In particular we extend the definition of NTR processes from the real line to classes of distributions on general spaces. Representations of the posterior distributions are given using a different type of calculus than traditionally used in the Bayesian literature. The techniques are applied to progressively more complex models. Refinements are then given which describes the underlying properties of spatial NTR models analogous to those developed for the Dirihclet process. The analysis yields accessible moment formulae and characterizations of the the posterior distribution and relavant marginal distributions. In the homogeneous case this work turns out to be connected to and overlap with recent work on regenerative compositions defined by a suitable discretisation of subordinators. The results also have connections to other related work on exponential functionals of subordinators. In addition, we develop results for spatial NTR mixture models and identify a class of species sampling models derived from spatial NTR processes. http://front.math.ucdavis.edu/math.ST/0305053 --------------------------------------------------------------- 3028. ON A MULTIVARIATE VERSION OF BERNSTEIN'S INEQUALITY P. Major We prove a multivariate version of Bernstein's inequality about the probability that degenerate $U$-statistics take a value larger than some number $u$. This is an improvement of former estimates for the same problem which yields an asymptotically sharp estimate for not too large numbers $u$. This paper also contains an analogous bound about the distribution of multiple Wiener-Ito integrals. Their comparison shows that our results are sharp. The proofs are based on good estimates about high moments of multiple random integrals. They are obtained by means of a diagram formula which enables us to express the product of multiple random integrals as the sum of such expressions. http://front.math.ucdavis.edu/math.PR/0411287 --------------------------------------------------------------- 3029. A MULTIVARIATE GENERALIZATION OF HOEFFDING'S INEQUALITY P. Major We prove a multivariate version of Hoeffding's inequality about the distribution of homogeneous polynomials of Rademacher functions. The proof is based on such an estimate about the moments of homogeneous polynomials of Rademacher functions which can be considered as an improvement of Borell's inequality in a most important special case. http://front.math.ucdavis.edu/math.PR/0411288 --------------------------------------------------------------- 3030. ALPHA-PFAFFIAN, PFAFFIAN POINT PROCESS AND SHIFTED SCHUR MEASURE Sho Matsumoto For any complex number $\alpha$ and any even-size skew-symmetric matrix $B$, we define a generalization $\pfa{\alpha}(B)$ of the pfaffian $\pf(B)$ which we call the $\alpha$-pfaffian. The $\alpha$-pfaffian is a pfaffian analogue of the $\alpha$-determinant. It gives the pfaffian at $\alpha=-1$. We give some formulas for $\alpha$-pfaffians and study the positivity. Further we define point processes determined by the $\alpha$-pfaffian. Also we provide a linear algebraic proof of the explicit pfaffian expression for the correlation function of the shifted Schur measure. http://front.math.ucdavis.edu/math.CO/0411277 --------------------------------------------------------------- 3031. GUNDY'S DECOMPOSITION FOR NON-COMMUTATIVE MARTINGALES AND APPLICATIONS Javier Parcet and Narcisse Randrianantoanina We provide an analogue of Gundy's decomposition for L1-bounded non-commutative martingales. An important difference from the classical case is that for any L1-bounded non-commutative martingale, the decomposition consists of four martingales. This is strongly related with the row/column nature of non-commutative Hardy spaces of martingales. As applications, we obtain simpler proofs of the weak type (1,1) boundedness for non-commutative martingale transforms and the non-commutative analogue of Burkholder's weak type inequality for square functions. A sequence (x_n) in a normed space X is called 2-co-lacunary if there exists a bounded linear map from the closed linear span of (x_n) to l2 taking each x_n to the n-th vector basis of l2. We prove (using our decomposition) that any relatively weakly compact martingale difference sequence in L1(M,\tau) whose sequence of norms is bounded away from zero is 2-co-lacunary, generalizing a result of Aldous and Fremlin to non-commutative L1-spaces. http://front.math.ucdavis.edu/math.OA/0411296 --------------------------------------------------------------- 3032. ON THE RECONSTRUCTION OF THE DRIFT OF A DIFFUSION FROM TRANSITION PROBABILITIES WHICH ARE PARTIALLY OBSERVED IN SPACE Sergio Albeverio and Carlo Marinelli The problem of reconstructing the drift of a diffusion in $\erre^d$, $d\geq 2$, from the transition probability density observed outside a domain is considered. The solution of this problem also solves a new inverse problem for a class of parabolic partial differential equations. This work considerably extends \cite{jsp} in terms of generality, both concerning assumptions on the drift coefficient, and allowing for non-constant diffusion coefficient. Sufficient conditions for solvability of this type of inverse problem for $d=1$ are also given. http://front.math.ucdavis.edu/math.PR/0411008 --------------------------------------------------------------- 3033. SOME REMARKS ON COMMUTATION RELATIONS FOR SLE Julien Dubedat Schramm-Loewner Evolutions (SLEs) describe a one-parameter family of growth processes in the plane that have particular conformal invariance properties. For instance, SLE can define simple random curves in a simply conneccted domain. In this paper we are interested in questions pertaining to the definition of several SLEs in a domain (i.e. several random curves). In particular, one derives infinitesimal commutation conditions, discuss some solutions, and show how to lift these infinitesimal relations to global relations in simple cases. http://front.math.ucdavis.edu/math.PR/0411299 --------------------------------------------------------------- 3034. ALMOST GLOBAL STOCHASTIC STABILITY Ramon van Handel We develop a method to prove almost global stability of stochastic differential equations in the sense that almost every initial point (with respect to the Lebesgue measure) is asymptotically attracted to the origin with unit probability. The method can be viewed as a dual to Lyapunov's second method for stochastic differential equations and extends the deterministic result in [A. Rantzer, Syst. Contr. Lett., 42 (2001), pp. 161--168]. The result can also be used in certain cases to find stabilizing controllers for stochastic nonlinear systems using convex optimization. The main technical tool is the theory of stochastic flows of diffeomorphisms. http://front.math.ucdavis.edu/math.PR/0411311 --------------------------------------------------------------- 3035. THE EMPIRICAL DISTRIBUTION OF THE EIGENVALUES OF A GRAM MATRIX WITH A GIVEN VARIANCE PROFILE W. Hachem and P. Loubaton and J. Najim Consider a $N\times n$ random matrix $Y_n=(Y_{ij}^{n})$ where the entries are given by $Y_{ij}^{n}=\frac{\sigma(i/N,j/n)}{\sqrt{n}} X_{ij}^{n}$, the $X_{ij}^{n}$ being centered i.i.d. and $\sigma:[0,1]^2 \to (0,\infty)$ being a continuous function called a variance profile. Consider now a deterministic $N\times n$ matrix $\Lambda_n=(\Lambda_{ij}^{n})$ whose non diagonal elements are zero. Denote by $\Sigma_n$ the non-centered matrix $Y_n + \Lambda_n$. Then under the assumption that $\lim_{n\to \infty} \frac Nn =c>0$ and $$ \frac{1}{N} \sum_{i=1}^{N} \delta_{(\frac{i}{N}, (\Lambda_{ii}^n)^2)} \xrightarrow[n\to \infty]{} H(dx,d\lambda), $$ where $H$ is a probability measure, it is proven that the empirical distribution of the eigenvalues of $ \Sigma_n \Sigma_n^T$ converges almost surely in distribution to a non random probability measure. This measure is characterized in terms of its Stieltjes transform, which is obtained with the help of an auxiliary system of equations. This kind of results is of interest in the field of wireless communication. http://front.math.ucdavis.edu/math.PR/0411333 --------------------------------------------------------------- 3036. DEVIATIONS OF A RANDOM WALK IN A RANDOM SCENERY WITH STRETCHED EXPONENTIAL TAILS Remco van der Hofstad and Nina Gantert and Wolfgang K{\"o}nig Let $(Z_n)_{n\in\N_0}$ be a d-dimensional random walk in random scenery, i.e., $Z_n=\sum_{k=0}^{n-1}Y_{S_k}$ with $(S_k)_{k\in\N_0}$ a random walk in $\Z^d$ and $(Y_{z})_{z\in\Z^d}$ an i.i.d. scenery, independent of the walk. We assume that the random variables $Y_{z}$ have a stretched exponential tail. In particular, they do not possess exponential moments. We identify the speed and the rate of the logarithmic decay of $\P(\frac 1n Z_n>t_n)$ for all sequences $(t_n)_{n\in\N}$ satisfying a certain lower bound. This complements results of \cite{GKS04}, where it was assumed that $Y_{z}$ has exponential moments of all orders. Informally, in contrast to the situation \cite{GKS04}, the event $\{\frac 1n Z_n>t_n\}$ is not realized by a homogeneous behavior of the walk's local times and the scenery, but by many visits of the walker to a particular site and a large value of the scenery at that site. This reflects a well-known extreme behavior typical for random variables having no exponential moments. http://front.math.ucdavis.edu/math.PR/0411361 --------------------------------------------------------------- 3037. AN EXTREME-VALUE ANALYSIS OF THE LIL FOR BROWNIAN MOTION Davar Khoshnevisan and David A. Levin and Zhan Shi We present an extreme-value analysis of the classical law of the iterated logarithm (LIL) for Brownian motion. Our result can be viewed as a new improvement to the LIL. http://front.math.ucdavis.edu/math.PR/0411376 --------------------------------------------------------------- 3038. THE HYPERBOLIC GEOMETRY OF RANDOM TRANSPOSITIONS Nathanael Berestycki Make the set of permutations of $n$ objects into a graph $G_n$ by connecting two permutations that differ by one transposition, and let $\sigma_t$ be the continuous time simple random walk on this graph. In a previous paper, Berestycki and Durrett (2004) showed that the limiting behavior of the distance from the identity at time $cn/2$ has a phase transition at $c=1$. When $c<1$, it is asymptotically $cn/2$, while for $c>1$ it is $u(c)n$ with $u(c) < c/2$. Here we investigate some consequences of this result for the geometry of $G_n$. Our first result is that when we consider the sphere of radius $an$ centered at the origin, and pick two points independently according to the hitting distribution, then Gromov hyperbolicity breaks down at critical radius $a=1/4$. When $a<1/4$ the space is hyperbolic but also displays behavior that is much different from manifolds of negative curvature - it is shown that there are many geodesics that may travel much different paths to get to a point. We also show that the hitting distribution of the sphere of radius $an$ is asymptotically singular with respect to the uniform distribution. Finally, we prove that the qualitative behavior of the Gromov hyperbolicity persists if we pick points independently according to the uniform measure on the sphere of radius $an$. However, in this case, the critical radius is $a=1-\log 2$. http://front.math.ucdavis.edu/math.PR/0411011 --------------------------------------------------------------- 3039. LARGE DEVIATIONS FOR DIFFUSIONS WITH TIME PERIODIC DRIFT AND STOCHASTIC RESONANCE Samuel Herrmann and Peter Imkeller and Dierk Peithmann We consider potential type dynamical systems in finite dimensions with two meta-stable states. They are subject to two sources of perturbation: a slow external periodic perturbation of period $T$ and a small Gaussian random perturbation of intensity $\eps$, and therefore mathematically described as weakly time inhomogeneous diffusion processes. A system is in stochastic resonance provided the small noisy perturbation is tuned in such a way that its random trajectories follow the exterior periodic motion in an optimal fashion, i.e. for some optimal intensity $\eps(T)$. The physicists' favorite measures of quality of periodic tuning -- and thus stochastic resonance -- such as spectral power amplification or signal-to-noise ratio have proven to be defective. They are not robust w.r.t. effective model reduction, i.e. for the passage to a simplified finite state Markov chain model reducing the dynamics to a pure jumping between the meta-stable states of the original system. An entirely probabilistic notion of stochastic resonance based on the transition dynamics between the domains of attraction of the meta-stable states -- and thus failing to suffer from this robustness defect -- was proposed before in the context of one-dimensional diffusions. It is investigated for higher dimensional systems here, by using extensions and refinements of the Freidlin-Wentzell theory of large deviations for time homogeneous diffusions. Large deviation principles developed for weakly time inhomogeneous diffusions prove to be key tools for a treatment of the problem of diffusion exit from a domain and thus for the approach of stochastic resonance via transition probabilities between meta-stable sets. http://front.math.ucdavis.edu/math.PR/0411386 --------------------------------------------------------------- 3040. COSINE PRODUCTS, FOURIER TRANSFORMS, AND RANDOM SUMS Kent E. Morrison We investigate several infinite product of cosines and find the closed form using the Fourier transform. The answers provide limiting distributions for some elementary probability experiments. http://front.math.ucdavis.edu/math.CA/0411380 --------------------------------------------------------------- 3041. POLYNUCLEAR GROWTH MODEL, GOE$^2$ AND RANDOM MATRIX WITH DETERMINISTIC SOURCE T. Imamura and T. Sasamoto We present a random matrix interpretation of the distribution functions which have appeared in the study of the one-dimensional polynuclear growth (PNG) model with external sources. It is shown that the distribution, GOE$^2$, which is defined as the square of the GOE Tracy-Widom distribution, can be obtained as the scaled largest eigenvalue distribution of a special case of a random matrix model with a deterministic source, which have been studied in a different context previously. Compared to the original interpretation of the GOE$^2$ as ``the square of GOE'', ours has an advantage that it can also describe the transition from the GUE Tracy-Widom distribution to the GOE$^2$. We further demonstrate that our random matrix interpretation can be obtained naturally by noting the similarity of the topology between a certain non-colliding Brownian motion model and the multi-layer PNG model with an external source. This provides us with a multi-matrix model interpretation of the multi-point height distributions of the PNG model with an external source. http://front.math.ucdavis.edu/math-ph/0411057 --------------------------------------------------------------- 3042. DOBRUSHIN-KOTECKY-SCHLOSMAN THEOREM FOR POLYGONAL MARKOV FIELDS IN THE PLANE Tomasz Schreiber We establish a version of the Dobrushin-Kotecky-Schlosman phase separation theorem for the length-interacting Arak-Surgailis polygonal Markov fields with V-shaped nodes. http://front.math.ucdavis.edu/math-ph/0411064 --------------------------------------------------------------- 3043. DISTRIBUTION FUNCTIONS FOR EDGE EIGENVALUES IN ORTHOGONAL AND SYMPLECTIC Momar Dieng We derive Painlev\'e--type expressions for the distribution of the $m^{th}$ largest eigenvalue in the Gaussian Orthogonal and Symplectic Ensembles in the edge scaling limit. The work of Johnstone and Soshnikov (see [7], [10]) implies the immediate relevance of our formulas for the $m^{th}$ largest eigenvalue of the appropriate Wishart distribution. http://front.math.ucdavis.edu/math.PR/0411421 --------------------------------------------------------------- 3044. QUANTUM HELE-SHAW FLOW Haakan Hedenmalm and Nikolai Makarov In this note, we discuss the quantum Hele-Shaw flow, a random measure process in the complex plane introduced by the physicists P.Wiegmann, A. Zabrodin, et al. This process arises in the theory of electronic droplets confined to a plane under a strong magnetic field, as well as in the theory of random normal matrices. We extend a result of Elbau and Felder to general external field potentials, and also show that if the potential is $C^2$-smooth, then the quantum Hele-Shaw flow converges, under appropriate scaling, to the classical (weighted) Hele-Shaw flow, which can be modeled in terms of an obstacle problem. http://front.math.ucdavis.edu/math.PR/0411437 --------------------------------------------------------------- 3045. FISHER'S INFORMATION FOR DISCRETELY SAMPLED LEVY PROCESSES Yacine Ait-Sahalia and Jean Jacod (PMA) This paper studies the asymptotic behavior of the Fisher information for a Levy process discretely sampled at an increasing frequency. We show that it is possible to distinguish not only the continuous part of the process from its jumps part, but also different types of jumps, and derive the rates of convergence of efficient estimators. http://front.math.ucdavis.edu/math.PR/0411438 --------------------------------------------------------------- 3046. GREEDY LATTICE ANIMALS: GEOMETRY AND CRITICALITY (WITH AN APPENDIX) Alan Hammond Assign to each site of the integer lattice $\Zd$ a real score, sampled according to the same distribution $F$, independently of the choices made at all other sites. A lattice animal is a finite connected set of sites, with its weight being the sum of the scores at its sites. Let $N_n$ be the maximal weight of those lattice animals of size $n$ that contain the origin. Denote by $N$ the almost sure finite constant limit of $n^{-1} N_n$, which exists under a mild condition on the positive tail of $F$. We study certain geometrical aspects of the lattice animal with maximal weight among those contained in an $n$-box where $n$ is large, both in the supercritical phase where $N > 0$, and in the critical case where $N = 0$. http://front.math.ucdavis.edu/math.PR/0411459 --------------------------------------------------------------- 3047. THE LARGEST EIGENVALUE OF SMALL RANK PERTURBATIONS OF HERMITIAN RANDOM MATRICES Sandrine P\'ech\'e We compute the limiting eigenvalue statistics at the edge of the spectrum of large Hermitian random matrices perturbed by the addition of small rank deterministic matrices. To be more precise, we consider random Hermitian matrices with independent Gaussian entries $M_{ij}, i\leq j$ with various expectations. We prove that the largest eigenvalue of such random matrices exhibits, in the large $N$ limit, various limiting distributions depending on both the eigenvalues of the matrix $(\mathbb{E}M_{ij})_{i,j=1}^N$ and its rank. http://front.math.ucdavis.edu/math.PR/0411487 --------------------------------------------------------------- 3048. STATISTICALLY DUAL DISTRIBUTIONS AND ESTIMATION OF THE PARAMETERS S.I. Bityukov and V.V. Smirnova and V.A. Taperechkina The reconstruction of the parameter of the model by the measurement of the random variable depending on this parameter is one of the main tasks of statistics. In the paper the notion of the statistically dual distributions is introduced. The approach, based on the properties of the statistically dual distributions, to resolving of the given task is proposed. http://front.math.ucdavis.edu/math.ST/0411462 --------------------------------------------------------------- 3049. SPONTANEOUS DYNAMICS OF ASYMMETRIC RANDOM RECURRENT SPIKING NEURAL NETWORKS H. Soula and G. Beslon and O. Mazet We study in this paper the effect of an unique initial stimulation on random recurrent networks of leaky integrate and fire neurons. Indeed given a stochastic connectivity this so-called spontaneous mode exhibits various non trivial dynamics. This study brings forward a mathematical formalism that allows us to examine the variability of the afterward dynamics according to the parameters of the weight distribution. Provided independence hypothesis (e.g. in the case of very large networks) we are able to compute the average number of neurons that fire at a given time -- the spiking activity. In accordance with numerical simulations, we prove that this spiking activity reaches a steady-state, we characterize this steady-state and explore the transients. http://front.math.ucdavis.edu/cs.NE/0411052 --------------------------------------------------------------- 3050. FAST NON-PARAMETRIC BAYESIAN INFERENCE ON INFINITE TREES Marcus Hutter Given i.i.d. data from an unknown distribution, we consider the problem of predicting future items. An adaptive way to estimate the probability density is to recursively subdivide the domain to an appropriate data-dependent granularity. A Bayesian would assign a data-independent prior probability to "subdivide", which leads to a prior over infinite(ly many) trees. We derive an exact, fast, and simple inference algorithm for such a prior, for the data evidence, the predictive distribution, the effective model dimension, and other quantities. http://front.math.ucdavis.edu/math.ST/0411515 --------------------------------------------------------------- 3051. EXCHANGEABLE PAIRS AND POISSON APPROXIMATION Sourav Chatterjee and Persi Diaconis and Elizabeth Meckes This is a survery paper on Poisson approximation using Stein's method of exchangeable pairs. We illustrate using Poisson-binomial trials and many variations on three classical problems of combinatorial probability: the matching problem, the coupon collector's problem, and the birthday problem. While many details are new, the results are closely related to a body of work developed by Andrew Barbour, Louis Chen, Richard Arratia, Lou Gordon, Larry Goldstein, and their collaborators. Some comparison with these other approaches is offered. http://front.math.ucdavis.edu/math.PR/0411525 --------------------------------------------------------------- 3052. FLUCTUATION OF PLANAR BROWNIAN LOOP CAPTURING LARGE AREA Alan Hammond and Yuval Peres We consider a planar Brownian loop $B$ that is run for a time $T$ and conditioned on the event that its range encloses the unusually high area of $\pi T^2$, with $T$ being large. We study the deviation of the range of the conditioned process $X$ from a circle of radius $T$, as a model for the fluctuation of a phase boundary. This deviation is measured by means of the inradius and outradius of the region enclosed by the range of $X$. We prove that in a typical realization of the conditioned measure, each of these quantities differs from $T$ by at most $T^{2/3 + \epsilon}$. http://front.math.ucdavis.edu/math.PR/0411540 --------------------------------------------------------------- 3053. THE INITIAL DRIFT OF A 2D DROPLET AT ZERO TEMPERATURE Raphael Cerf and Sana Louhichi We consider the 2D stochastic Ising model evolving according to the Glauber dynamics at zero temperature. We compute the initial drift for droplets which are discretizations of smooth domains. A specific spatial average of the derivative at time~0 of the volume variation of a droplet close to a boundary point is equal to its curvature multiplied by a direction dependent coefficient. For a boundary point having a tangent with angle $\theta$, this coefficient is equal to $-\frac{\textstyle 1}{\textstyle 2}|\cos 2\theta|$. http://front.math.ucdavis.edu/math.PR/0411545 --------------------------------------------------------------- 3054. THE EFFECT OF FINITE MEMORY CUTOFF ON LOOP ERASED WALK IN Z^3 Wei-Shih Yang and Aklilu Zeleke Let \zeta be the intersection exponent of random walks in Z^3 and \alpha be a positive real number. We construct a stochastic process from a simple random walk by erasing loops of length at most N^\alpha. We will prove that for \alpha < \frac{1}{1+2\zeta}, the limiting distribution is Gaussian. For \alpha > 2 the limiting distribution will be shown to be equal to the limiting distribution of the loop erased walk. http://front.math.ucdavis.edu/math.PR/0411551 --------------------------------------------------------------- 3055. EXACT VARIATIONS FOR STOCHASTIC HEAT EQUATIONS DRIVEN BY SPACE--TIME WHITE NOISE Jan Pospisil and Roger Tribe This paper calculates the exact quadratic variation in space and quartic variation in time for the solutions to a one dimensional stochastic heat equation driven by a multiplicative space-time white noise. http://front.math.ucdavis.edu/math.PR/0411552 --------------------------------------------------------------- 3056. COMPACTNESS OF THE LIMIT SET IN FIRST-PASSAGE PERCOLATION ON VORONOI TILINGS Leandro P.R. Pimentel In this paper we consider first-passage percolation models on Voronoi tilings of the plane and present a sufficient condition for the compactness of the limit set. This result is based on a static renormalization technique and also provide an inequality involving critical probabilities for bond percolation models. http://front.math.ucdavis.edu/math.PR/0411560 --------------------------------------------------------------- 3057. COMPETING GROWTH AND GEODESICS IN FIRST-PASSAGE PERCOLATION Leandro P.R. Pimentel We consider a competing spatial growth dynamics permitting that more than one cluster develop in the same environment given by a first-passage percolation model on a Voronoi tiling of the plane. We focus on the long time behavior of these competing clusters and derive some limit theorems related to the morphology of the ``competition interface''. To study the structure of this interface we use the notion of geodesic in first-passage percolation and explore the coalescence behavior of semi-infinite geodesics with the same orientation. http://front.math.ucdavis.edu/math.PR/0411583 --------------------------------------------------------------- 3058. A SHARP ISOPERIMETRIC BOUND FOR CONVEX BODIES Ravi Montenegro We consider the problem of lower bounding a generalized Minkowski measure of subsets of a convex body with a log-concave probability measure, conditioned on the set size. A bound is given in terms of diameter and set size, which is sharp for all set sizes, dimensions, and norms. In the case of uniform density a stronger theorem is shown which is also sharp. http://front.math.ucdavis.edu/math.FA/0411018 --------------------------------------------------------------- 3059. ON EXPONENTIAL STABILITY OF THE NONLINEAR FILTER FOR SLOWLY SWITCHING MARKOV CHAINS P. Chigansky Exponential stability of the nonlinear filtering recursion is revisited, when the signal is a finite state Markov chain. An asymptotic upper bound for the filtering error due to incorrect initial condition is derived for the case of slowly switching signal. http://front.math.ucdavis.edu/math.PR/0411596 --------------------------------------------------------------- 3060. THE QUANTIZATION COMPLEXITY OF DIFFUSION PROCESSES Steffen Dereich We investigate the high resolution coding problem for solutions of stochastic differential equations in the L^p[0,1]- and the C[0,1]-space. Tight asymptotic estimates are found under weak regularity assumptions. The main technical tool is a decoupling method which allows us to relate the complexity of the diffusion process to that of the Wiener process under certain random distortions. http://front.math.ucdavis.edu/math.PR/0411597 --------------------------------------------------------------- 3061. AN ALMOST SURE INVARIANCE PRINCIPLE FOR RANDOM WALKS IN A SPACE-TIME RANDOM ENVIRONMENT F. Rassoul-Agha and T. Seppalainen We consider a discrete time random walk in a space-time i.i.d. random environment. We use a martingale approach to show that the walk is diffusive in almost every fixed environment. We improve on existing results by proving an invariance principle and considering environments with an annealed $L^2$ drift. We also state an a.s. invariance principle for random walks in general random environments whose hypothesis requires a subdiffusive bound on the variance of the quenched mean, under an ergodic invariant measure for the environment chain. http://front.math.ucdavis.edu/math.PR/0411602 --------------------------------------------------------------- 3062. AN ALMOST SURE INVARIANCE PRINCIPLE FOR ADDITIVE FUNCTIONALS OF MARKOV CHAINS F. Rassoul-Agha and T. Seppalainen We prove an invariance principle for a vector-valued additive functional of a Markov chain for almost every starting point with respect to an ergodic equilibrium distribution. The hypothesis is a moment bound on the resolvent. http://front.math.ucdavis.edu/math.PR/0411603 --------------------------------------------------------------- 3063. NONCOMMUTATIVE CONTINUOUS BERNOULLI SHIFTS J\"urgen Hellmich and Claus K\"ostler and Burkhard K\"ummerer We introduce a non-commutative extension of Tsirelson-Vershik's noises, called (non-commutative) continuous Bernoulli shifts. These shifts encode stochastic independence in terms of commuting squares, as they are familiar in subfactor theory. Such shifts are, in particular, capable of producing Arveson's product system of type I and type II. We investigate the structure of these shifts and prove that the von Neumann algebra of a (scalar-expected) continuous Bernoulli shift is either finite or of type III. The role of (`classical') stationary flows for Tsirelson-Vershik's noises is now played by cocycles of continuous Bernoulli shifts. We show that these cocycles provide an operator algebraic notion for Levy processes. They lead, in particular, to units and `logarithms' of units in Arveson's product systems. Furthermore, we introduce (non-commutative) white noises, which are operator algebraic versions of Tsirelson's `classical' noises. We give examples coming from probability, quantum probability and from Voiculescu's theory of free probability. Our main result is a bijective correspondence between additive and unital shift cocycles. For the proof of the correspondence we develop tools which are of interest on their own: non-commutative extensions of stochastic Ito integration, stochastic logarithms and exponentials. http://front.math.ucdavis.edu/math.OA/0411565 --------------------------------------------------------------- 3064. EXACT CONSTANTS IN THE ROSENTHAL MOMENT INEQUALITIES FOR SUMS OF INDEPENDENT CENTERED RANDOM VARIABLES B. Naimark and E. Ostrovsky We study the exact constants in the moment inequalities for sums of centered independent random variables: improve their asymptotics, low and upper bounds, calculate more exact asymptotics, elaborate the numerical algorithm for their calculation, study the class of smoothing etc. http://front.math.ucdavis.edu/math.PR/0411614 --------------------------------------------------------------- 3065. EXPONENTIAL BOUNDS FOR RANDOM SUMS B.M. Migdashiev and E.I. Ostrovsky We construct a non - improved exponential bounds for distribution of normed sums of i.,i.d. random variables with random numbers of summand. http://front.math.ucdavis.edu/math.PR/0411616 --------------------------------------------------------------- 3066. LONG RANGE EXCLUSION PROCESSES, GENERATOR AND INVARIANT MEASURES E. D. Andjel (Univ. Provence) and H. Guiol (INP Grenoble) We show that if $\mu$ is an invariant measure for the long range exclusion process putting no mass on the full configuration, $L$ is the formal generator of that process, and $f$ is a cylinder function, then $Lf \in\mathbf{L}^{1} (d\mu)$ and $\int Lf d\mu =0$. This result is then applied to determine, i) the set of invariant and translation invariant measures of the long range exclusion process on $\Z^d$ when the underlying random walk is irreducible; ii) the set of invariant measures of the long range exclusion process on $\Z$ when the underlying random walk is irreducible and has either zero mean or allows jumps only to the nearest neighbors. http://front.math.ucdavis.edu/math.PR/0411655 --------------------------------------------------------------- 3067. LARGE DEVIATIONS FOR TRAPPED INTERACTING BROWNIAN PARTICLES AND PATHS Stefan Adams and Jean-Bernard Bru and Wolfgang Koenig We introduce two probabilistic models for $N$ interacting Brownian motions moving in a trap in $ \R^d $ under mutually repellent forces. The two models are defined in terms of transformed path measures on finite time intervals under a trap Hamiltonian and two respective pair-interaction Hamiltonians. The first pair interaction exhibits a {\it particle} repellency, while the second one imposes a {\it path} repellency. We analyse both models in the limit of diverging time with fixed number $ N $ of Brownian motions. In particular, we prove large deviations principles for the normalised occupation measures. The minimisers of the rate functions are related to the Hamilton operator for $ N $ interacting trapped particles. More precisely, in the particle-repellency model, the minimiser is its ground state, and in the path-repellency model, the minimisers are its ground product-states. This study is a contribution to the search for a mathematical formulation of the quantum system of $ N $ trapped interacting bosons as a model for {\it Bose-Einstein condensation}, motivated by the success of the famous 1995 experiments. Recently, Lieb, et al. described the large-N behaviour of the ground state in terms of the well-known {\it Gross-Pitaevskii} formula, involving the scattering length of the pair potential. We prove that the large-N behaviour of the ground product-states is also described by the Gross-Pitaevskii formula, however with the scattering length of the pair potential replaced by its integral. http://front.math.ucdavis.edu/math.PR/0411660 --------------------------------------------------------------- 3068. DEFINITION OF A DETERMINISTIC BAYESIAN LOGIC Frederic Dambreville (DGA/CTA/DT/GIP) The Bayesian logic is generally associated to the definition of a prior probabilistic law. Conditional algebra have been investigated by some authors though, but somehow the background framework is still probabilistic and the entire logic is not specified. In this paper, the definition of a Deterministic Bayesian Logic is proposed. This logic is completely independent of any notion of probability. The coherence of this logic is proven and various logical theorems are derived. It is shown that this logic is probabilizable and avoids the negative result of Lewis. At last the probabilistic Bayesian rule is recovered by posteriorly probabilizing our logic. http://front.math.ucdavis.edu/cs.LO/0411097 --------------------------------------------------------------- 3069. THE PEARCEY PROCESS Craig A. Tracy and Harold Widom The extended Airy kernel describes the space-time correlation functions for the Airy process, which is the limiting process for a polynuclear growth model. The Airy functions themselves are given by integrals in which the exponents have a cubic singularity, arising from the coalescence of two saddle points in an asymptotic analysis. Pearcey functions are given by integrals in which the exponents have a quartic singularity, arising from the coalescence of three saddle points. A corresponding Pearcey kernel appears in a random matrix model and a Brownian motion model for a fix time. This paper derives an extended Pearcey kernel by scaling the Brownian motion model at several times, and a system of partial differential equations whose solution determines associated distribution functions. We expect there to be a limiting nonstationary process consisting of infinitely many paths, which we call the Pearcey process, whose space-time correlation functions are expressible in terms of this extended kernel. http://front.math.ucdavis.edu/math.PR/0412005 --------------------------------------------------------------- 3070. FREQUENTLY VISITED SETS FOR RANDOM WALKS Endre Cs\'aki and Ant\'onia F\"oldes and P\'al R\'ev\'esz and Jay Rosen and Zhan Shi We study the occupation measure of various sets for a symmetric transient random walk in $Z^d$ with finite variances. Let $\mu^X_n(A)$ denote the occupation time of the set $A$ up to time $n$. It is shown that $\sup_{x\in Z^d}\mu_n^X(x+A)/\log n$ tends to a finite limit as $n\to\infty$. The limit is expressed in terms of the largest eigenvalue of a matrix involving the Green's function of $X$ restricted to the set $A$. Some examples are discussed and the connection to similar results for Brownian motion is given. http://front.math.ucdavis.edu/math.PR/0412018 --------------------------------------------------------------- 3071. CONVEX IMPRECISE PREVISIONS: BASIC ISSUES AND APPLICATIONS Renato Pelessoni and Paolo Vicig In this paper we study two classes of imprecise previsions, which we termed convex and centered convex previsions, in the framework of Walley's theory of imprecise previsions. We show that convex previsions are related with a concept of convex natural estension, which is useful in correcting a large class of inconsistent imprecise probability assessments. This class is characterised by a condition of avoiding unbounded sure loss. Convexity further provides a conceptual framework for some uncertainty models and devices, like unnormalised supremum preserving functions. Centered convex previsions are intermediate between coherent previsions and previsions avoiding sure loss, and their not requiring positive homogeneity is a relevant feature for potential applications. Finally, we show how these concepts can be applied in (financial) risk measurement. http://front.math.ucdavis.edu/math.PR/0412030 --------------------------------------------------------------- 3072. A PROBABILISTIC REPRESENTATION OF SOLUTIONS OF THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS IN R3 M. Ossiander A new probabilistic representation is presented for solutions of the incompressible Navier-Stokes equations in 3 dimensions with given forcing and initial velocity. This representation expresses solutions as scaled conditional expectations of functionals of a Markov process indexed by the nodes of a binary tree. It gives existence and uniqueness of weak solutions for all time under relatively simple conditions on the forcing and initial data. These conditions involve comparison of the forcing and initial data with majorizing kernels. http://front.math.ucdavis.edu/math.PR/0412034 --------------------------------------------------------------- 3073. BENFORD'S LAW, VALUES OF L-FUNCTIONS AND THE 3X+1 PROBLEM Alex V. Kontorovich and Steven J. Miller We show the leading digits of a variety of systems satisfying certain conditions follow Benford's Law. For each system proving this involves two main ingredients. One is a structure theorem of the limiting distribution, specific to the system. The other is a general technique of applying Poisson Summation to the limiting distribution. We show the distribution of values of L-functions near the central line and (in some sense) the iterates of the 3x+1 Problem are Benford. http://front.math.ucdavis.edu/math.NT/0412003