Probability Abstracts 96

This document contains abstracts 5093-5304 from Jan-1-2007 to Feb-28-2007.
They have been mailed on March 1st, 2007.

5093. Expected Number of Slope Crossings of Certain Gaussian Random Polynomials

Author(s): S. Rezakhah and S. Shemehsavar

Abstract: Let $Q_n(x)=\sum_{i=0}^{n} A_{i}x^{i}$ be a random polynomial where the coefficients $A_0,A_1,... $ form a sequence of centered Gaussian random variables. Moreover, assume that the increments $\Delta_j=A_j-A_{j-1}$, $j=0,1,2,...$ are independent, assuming $A_{-1}=0$. The coefficients can be considered as $n$ consecutive observations of a Brownian motion. We study the number of times that such a random polynomial crosses a line which is not necessarily parallel to the x-axis. More precisely we obtain the asymptotic behavior of the expected number of real roots of the equation $Q_n(x)=Kx$, for the cases that $K$ is any non-zero real constant $K=o(n^{1/4})$, and $K=o(n^{1/2})$ separately.

http://arXiv.org/abs/math/0701019
http://front.math.ucdavis.edu/math.PR/0701019 (alternate)

5094. Tilted stable subordinators, Gamma time changes and Occupation Time of rays by Bessel Spiders

Author(s): Lancelot F. James and Marc Yor

Abstract: We exhibit, in the form of some identities in law, some connections between tilted stable subordinators, time-changed by independent Gamma processes and the occupation times of Bessel spiders, or their bridges. These identities in law are then explained thanks to excursion theory.

http://arXiv.org/abs/math/0701049
http://front.math.ucdavis.edu/math.PR/0701049 (alternate)

5095. Intractability rate of approximation problem for random fields in increasing dimension

Author(s): N. Serdyukova

Abstract: The behavior of average approximation cardinality for d-parametric random fields of tensor product type is investigated. The exact rate of dimension curse is obtained.

http://arXiv.org/abs/math/0701058
http://front.math.ucdavis.edu/math.PR/0701058 (alternate)

5096. Goodness of fit test for ergodic diffusion processes

Author(s): Ilia Negri and Yoichi Nishiyama

Abstract: A goodness of fit test for the drift coefficient of an ergodic diffusion process is presented. The test is based on the score marked empirical process. The weak convergence of the proposed test statistic is studied under the null hypotheses and it is proved that the limit process is a continuous Gaussian process. The structure of its covariance function allows to calculate the limit distribution and it turns out that it is a function of a standard Brownian motion and so exact reject regions can be constructed. The proposed test is asymptotically distribution free and it is consistent under any simple fixed alternative.

http://arXiv.org/abs/math/0701022
http://front.math.ucdavis.edu/math.ST/0701022 (alternate)

5097. A Law of Large Numbers for an Interacting Particle System with Confining Potential

Author(s): Matteo Ortisi (Dept. of Mathematics and University of Milano)

Abstract: In this paper we consider an interacting particle system modeled as a system of $N$ stochastic differential equations driven by Brownian motions with a drift term including a confining potential acting on each particle, and an interaction potential modeling the interaction among all the particles of the system. The limiting behavior as the size $N$ grows to infinity is achieved as a law of large numbers for the empirical process associated with the interacting particle system

http://arXiv.org/abs/math/0701095
http://front.math.ucdavis.edu/math.PR/0701095 (alternate)

5098. Compressed Sensing and Redundant Dictionaries

Author(s): Holger Rauhut and Karin Schnass and Pierre Vandergheynst

Abstract: This article extends the concept of compressed sensing to signals that are not sparse in an orthonormal basis but rather in a redundant dictionary. It is shown that a matrix, which is a composition of a random matrix of certain type and a deterministic dictionary, has small restricted isometry constants. Thus, signals that are sparse with respect to the dictionary can be recovered via Basis Pursuit from a small number of random measurements. Further, thresholding is investigated as recovery algorithm for compressed sensing and conditions are provided that guarantee reconstruction with high probability. The different schemes are compared by numerical experiments.

http://arXiv.org/abs/math/0701131
http://front.math.ucdavis.edu/math.PR/0701131 (alternate)

5099. Short-length routes in low-cost networks via Poisson line patterns

Author(s): David J. Aldous and Wilfrid S. Kendall

Abstract: In designing a network to link n cities in a square of area n, one might be guided by the following two desiderata. First, the total network length should not be much greater than the length of the shortest network connecting all cities. Second, the average route length (taken over source-destination pairs) should not be much greater than the average straight-line distance. How small can we make these two differences? For typical configurations the shortest network length is order n and the average straight-line distance is order n^1/2, so it seems implausible that one can construct a network in which the first difference is o(n) and the second difference is o(n^1/2). But in fact one can do better: for an arbitrary configuration one can construct a network where the first difference is o(n) and the second difference is almost as small as O(log n). The construction is conceptually simple: over the minimum-length connected network (Steiner tree) superimpose a sparse stationary and isotropic Poisson line process. The key ingredient is a new result about the Poisson line process. Consider two points at distance r apart, and delete from the line process all lines which separate these two points. The resulting pattern of lines partitions the plane into cells; the cell containing the two points has mean boundary length 2r + C log r. Turning to lower bounds we show that, under a weak equidistribution assumption, if the first difference is o(n) then the second difference cannot be O(sqrt(log n)).

http://arXiv.org/abs/math/0701140
http://front.math.ucdavis.edu/math.PR/0701140 (alternate)

5100. The Central Limit Theorem for LS Estimator in Simple Linear Ev Regression Models

Author(s): Yu Miao and Guangyu Yang and Luming Shen

Abstract: In this paper, we obtain the central limit theorems for LS estimator in simple linear errors-in-variables (EV) regression models under some mild conditions. And we also show that those conditions are necessary in some sense.

http://arXiv.org/abs/math/0701162
http://front.math.ucdavis.edu/math.PR/0701162 (alternate)

5101. The law of the iterated logarithm for additive functionals of Markov chains

Author(s): Yu Miao and Guangyu Yang

Abstract: In the paper, the law of the iterated logarithm for additive functionals of Markov chains is obtained under some weak conditions, which are weaker than the conditions of invariance principle of additive functionals of Markov chains in M. Maxwell and M. Woodroofe (2000). The main technique is the martingale argument and the theory of fractional coboundaries.

http://arXiv.org/abs/math/0701167
http://front.math.ucdavis.edu/math.PR/0701167 (alternate)

5102. One dimensional nearest neighbor exclusion processes in inhomogeneous and random environments

Author(s): Lincoln Chayes and Thomas M. Liggett

Abstract: The processes described in the title always have reversible stationary distributions. In this paper, we give sufficient conditions for the existence of, and for the nonexistence of, nonreversible stationary distributions. In the case of an i.i.d. environment, these combine to give a necessary and sufficient condition for the existence of nonreversible stationary distributions.

http://arXiv.org/abs/math/0701180
http://front.math.ucdavis.edu/math.PR/0701180 (alternate)

5103. A note for extension of almost sure central limit theory

Author(s): Yu Miao and Guangyu Yang

Abstract: H\"ormann (2006) gave an extension of almost sure central limit theorem for bounded Lipschitz 1 function. In this paper, we show that his result of almost sure central limit theorem is also hold for any Lipschitz function under stronger conditions.

http://arXiv.org/abs/math/0701183
http://front.math.ucdavis.edu/math.PR/0701183 (alternate)

5104. On a type Sobolev inequality and its applications

Author(s): Witold Bednorz

Abstract: In the paper we pursue the analysis from the section 5 of the Talagrand's paper "Sample boundedness of stochastic processes under increment conditions." Ann. Probab. 18, No. 1, 1-49. In particular we give the proof of some Sobolev Inequality and then apply it to obtain if and only if condition for all processes with bounded icrements to have bounded samples. The processes are defined on a compact, concave subspaces of $\R^n$ with a metric $d(s,t)=\eta(||s-t||)$, where $\eta$ is concave and $||.||$ is a norm on $\R^n$.

http://arXiv.org/abs/math/0701191
http://front.math.ucdavis.edu/math.PR/0701191 (alternate)

5105. Diffusion Limited Aggregation on a Cylinder

Author(s): Itai Benjamini and Ariel Yadin

Abstract: We consider the DLA process on a cylinder $G \times \N$. It is shown that this process ``grows arms'', provided that the base graph $G$ has small enough mixing time. Specifically, if the mixing time of $G$ is at most $\log^{(2-\eps)}\abs{G}$, the time it takes the cluster to reach the $m$-th layer of the cylinder is at most of order $m \cdot \frac{\abs{G}}{\log\log \abs{G}}$. In particular we get examples of infinite Cayley graphs of degree 5, for which the DLA cluster on these graphs has arbitrarily small density. In addition, we provide an upper bound on the rate at which the ``arms'' grow. This bound is valid for a large class of base graphs $G$, including discrete tori of dimension at least 3. It is also shown that for any base graph $G$, the density of the DLA process on a $G$-cylinder is related to the rate at which the arms of the cluster grow. This implies, that for any vertex transitive $G$, the density of DLA on a $G$-cylinder is bounded by 2/3.

http://arXiv.org/abs/math/0701201
http://front.math.ucdavis.edu/math.PR/0701201 (alternate)

5106. On the constructions of the skew Brownian motion

Author(s): Antoine Lejay

Abstract: This article summarizes the various ways one may use to construct the Skew Brownian motion, and shows their connections. Recent applications of this process in modelling and numerical simulation motivates this survey. This article ends with a brief account of related results, extensions and applications of the Skew Brownian motion.

http://arXiv.org/abs/math/0701219
http://front.math.ucdavis.edu/math.PR/0701219 (alternate)

5107. A Markov chain model of a polling system with parameter regeneration

Author(s): Iain MacPhee and Mikhail Menshikov and Dimitri Petritis and and Serguei Popov

Abstract: We study a model of a polling system i.e. a collection of $d$ queues with a single server that switches from queue to queue. The service time distribution and arrival rates change randomly every time a queue is emptied. This model is mapped to a mathematically equivalent model of a random walk with random choice of transition probabilities, a model which is of independent interest. All our results are obtained using methods from the constructive theory of Markov chains. We determine conditions for the existence of polynomial moments of hitting times for the random walk. An unusual phenomenon of thickness of the region of null recurrence for both the random walk and the queueing model is also proved.

http://arXiv.org/abs/math/0701226
http://front.math.ucdavis.edu/math.PR/0701226 (alternate)

5108. Functional CLT for random walk among bounded random conductances

Author(s): Marek Biskup and Timothy M. Prescott

Abstract: We consider the nearest-neighbor simple random walk on $\Z^d$, $d\ge2$, driven by a field of i.i.d. random nearest-neighbor conductances $\omega_{xy}\in[0,1]$. Apart from the requirement that the bonds with positive conductances percolate, we pose no restriction on the law of the $\omega$'s. We prove that, for a.e. realization of the environment, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. This holds despite the fact that the local CLT may fail in $d\ge5$ due to anomalously slow decay of the probability that the walk returns to the starting point at a given time (cf math.PR/0611666).

http://arXiv.org/abs/math/0701248
http://front.math.ucdavis.edu/math.PR/0701248 (alternate)

5109. Diffusivity in one-dimensional generalized Mott variable-range hopping models

Author(s): Pietro Caputo and Alessandra Faggionato

Abstract: We consider random walks in random environment which are generalized versions of well known effective models for Mott variable--range hopping. We study the homogenized diffusion constant of the random walk in the one--dimensional case. We prove various estimates on the the low--temperature behavior which confirm and extend previous work by physicists.

http://arXiv.org/abs/math/0701253
http://front.math.ucdavis.edu/math.PR/0701253 (alternate)

5110. Precise logarithmic asymptotics for the right tails of some limit random variables for random trees

Author(s): James Allen Fill and Svante Janson

Abstract: For certain random variables that arise as limits of functionals of random finite trees, we obtain precise asymptotics for the logarithm of the right-hand tail. Our results are based on the facts (i) that the random variables we study can be represented as functionals of a Brownian excursion and (ii) that a large deviation principle with good rate function is known explicitly for Brownian excursion. Examples include limit distributions of the total path length and of the Wiener index in conditioned Galton-Watson trees (also known as simply generated trees). In the case of Wiener index (where we recover results proved by Svante Janson and Philippe Chassaing by a different method) and for some other examples, a key constant is expressed as the solution to a certain optimization problem, but the constant's precise value remains unknown.

http://arXiv.org/abs/math/0701259
http://front.math.ucdavis.edu/math.PR/0701259 (alternate)

5111. Fluctuations of Levy processes and scattering theory

Author(s): Sonia Fourati

Abstract: We establish a connection between the scattering inverse problem and the determination of the distribution of the position of the Levy process at the exit time of a bounded interval in term of its Levy exponent.

http://arXiv.org/abs/math/0701271
http://front.math.ucdavis.edu/math.PR/0701271 (alternate)

5112. An improvement of a result on Smolyanov-Weizsaecker surface measures

Author(s): Evelina Shamarova

Abstract: Let $M$ be a compact Riemannian manifold without boundary isometrically embedded into $\Rnu^m$, $\W^x_{M,t}$ be the distribution of a Brownian bridge starting at $x\in M$ and returning to $M$ at time $t$. Let $Q_t: \C(M) \to \C(M)$, $(Q_t f)(x)=\int_{\C([0,1],\Rnu^m)}f(\om(t)) \W^x_{M,t}(d\om)$, and let $\mc P = \{0=t_0 < t_1 < ... < t_n=t\}$ be a partition of $[0,t]$. It was shown in a paper by O. G. Smolyanov, H. v. Weizsaecker, and O. Wittich that $Q_{t_1-t_0}... Q_{t_n-t_{n-1}} f \to e^{-t\frac{\lap_M}2}f, \text{as} |\mc P|\to 0$ in $\C(M)$. Taking into consideration integral representations: $(Q_{t_1-t_0}... Q_{t_n-t_{n-1}} f)(x)=\int_M q_{_{\mc P}}(x,y)f(y)\la_M(dy)$ and $(e^{-t\frac{\lap_M}2}f)(x)=\int_M h(x,y,t) f(y) \la_M(dy)$, where $\la_M$ is the volume measure on $M$, $h(x,y,t)$ is the heat kernel on $M$, one interprets this relation as a weak convergence in $\C(M)$ of the integral kernels: $q_{\mc P}(x,y)\to h(x,y,t)$. The present paper improves the result by Smolyanov and Weizsaecker, and shows that this convergence is uniform on $M\x M$. Keywords: Gaussian integrals on compact Riemannian manifolds, heat kernel, Smolyanov--Weizsaecker approach, Smolyanov--Weizsaecker surface measures

http://arXiv.org/abs/math/0701281
http://front.math.ucdavis.edu/math.PR/0701281 (alternate)

5113. The convergence to equilibrium of neutral genetic models

Author(s): Pierre Del Moral (JAD and IRISA / INRIA Rennes) and Laurent Miclo (LATP) and Fr\'{e}d\'{e}ric Patras (JAD), Sylvain Rubenthaler (JAD)

Abstract: This article is concerned with the long time behavior of neutral genetic population models, with fixed population size. We design an explicit, finite, exact, genealogical tree based representation of stationary populations that holds both for finite and infinite types (or alleles) models. We then analyze the decays to the equilibrium of finite populations in terms of the convergence to stationarity of their first common ancestor. We estimate the Lyapunov exponent of the distribution flows with respect to the total variation norm. We give bounds on these exponents only depending on the stability with respect to mutation of a single individual; they are inversely proportional to the population size parameter.

http://arXiv.org/abs/math/0701284
http://front.math.ucdavis.edu/math.PR/0701284 (alternate)

5114. Sorting using complete subintervals and the maximum number of runs in a randomly evolving sequence

Author(s): Svante Janson

Abstract: We study the space requirements of a sorting algorithm where only items that at the end will be adjacent are kept together. This is equivalent to the following combinatorial problem: Consider a string of fixed length n that starts as a string of 0's, and then evolves by changing each 0 to 1, with then changes done in random order. What is the maximal number of runs of 1's? We give asymptotic results for the distribution and mean. It turns out that, as in many problems involving a maximum, the maximum is asymptotically normal, with fluctuations of order n^{1/2}, and to the first order well approximated by the number of runs at the instance when the expectation is maximized, in this case when half the elements have changed to 1; there is also a second order term of order n^{1/3}. We also treat some variations, including priority queues. The proofs use methods originally developed for random graphs.

http://arXiv.org/abs/math/0701288
http://front.math.ucdavis.edu/math.PR/0701288 (alternate)

5115. Critical random graphs: diameter and mixing time

Author(s): Asaf Nachmias and Yuval Peres

Abstract: Let C_1 denote the largest connected component of the critical Erdos-Renyi random graph G(n,1/n). We show that, typically, the diameter of C_1 is of order n^{1/3} and the mixing time of the lazy simple random walk on C_1 is of order n. The latter answers a question of Benjamini, Kozma and Wormald. These results extend to clusters of size n^{2/3} of p-bond percolation on any d-regular n-vertex graph where such clusters exist, provided that p(d-1) \leq 1.

http://arXiv.org/abs/math/0701316
http://front.math.ucdavis.edu/math.PR/0701316 (alternate)

5116. Asymptotic normality for traces of polynomials in independent complex Wishart matrices

Author(s): Wlodek Bryc

Abstract: We derive a non-asymptotic expression for the moments of traces of monomials in several independent complex Wishart matrices, extending some explicit formulas available in the literature. We then deduce the explicit expression for the cumulants. From the latter, we read out the multivariate normal approximation to the traces of finite families of polynomials in independent complex Wishart matrices.

http://arXiv.org/abs/math/0701318
http://front.math.ucdavis.edu/math.PR/0701318 (alternate)

5117. Some Theory for the Analysis of Random Fields - With Applications to Geostatistics

Author(s): Philipp Pluch

Abstract: MSc thesis written under the supervision of Dr. J. Pilz (Klagenfurt University) and Dr. W. Mueller (Linz University) during the FWF Project 'Optimal design of correlated random fields'.

http://arXiv.org/abs/math/0701323
http://front.math.ucdavis.edu/math.ST/0701323 (alternate)

5118. Random Matrices, the Ulam Problem, Directed Polymers & Growth Models, and Sequence Matching

Author(s): Satya N. Majumdar

Abstract: In these lecture notes I will give a pedagogical introduction to some common aspects of 4 different problems: (i) random matrices (ii) the longest increasing subsequence problem (also known as the Ulam problem) (iii) directed polymers in random medium and growth models in (1+1) dimensions and (iv) a problem on the alignment of a pair of random sequences. Each of these problems is almost entirely a sub-field by itself and here I will discuss only some specific aspects of each of them. These 4 problems have been studied almost independently for the past few decades, but only over the last few years a common thread was found to link all of them. In particular all of them share one common limiting probability distribution known as the Tracy-Widom distribution that describes the asymptotic probability distribution of the largest eigenvalue of a random matrix. I will mention here, without mathematical derivation, some of the beautiful results discovered in the past few years. Then, I will consider two specific models (a) a ballistic deposition growth model and (b) a model of sequence alignment known as the Bernoulli matching model and discuss, in some detail, how one derives exactly the Tracy-Widom law in these models. The emphasis of these lectures would be on how to map one model to another. Some open problems are discussed at the end.

http://arXiv.org/abs/cond-mat/0701193
http://front.math.ucdavis.edu/cond-mat/0701193 (alternate)

5119. Percolation on dense graph sequences

Author(s): B. Bollobas and C. Borgs and J. Chayes and O. Riordan

Abstract: In this paper, we determine the percolation threshold for an arbitrary sequence of dense graphs $(G_n)$. Let $\lambda_n$ be the largest eigenvalue of the adjacency matrix of $G_n$, and let $G_n(p_n)$ be the random subgraph of $G_n$ that is obtained by keeping each edge independently with probability $p_n$. We show that the appearance of a giant component in $G_n(p_n)$ has a sharp threshold at $p_n=1/\lambda_n$. In fact, we prove much more, that if $(G_n)$ converges to an irreducible limit, then the density of the largest component of $G_n(c/n)$ tends to the survival probability of a multi-type branching process defined in terms of this limit. Here the notions of convergence and limit are those of Borgs, Chayes, Lov\'asz, S\'os and Vesztergombi. In addition to using basic properties of convergence, we make heavy use of the methods of Bollob\'as, Janson and Riordan, who used such branching processes to study the emergence of a giant component in a very broad family of sparse inhomogeneous random graphs.

http://arXiv.org/abs/math/0701346
http://front.math.ucdavis.edu/math.PR/0701346 (alternate)

5120. A graph theoretic interpretation of the mean first passage times

Author(s): Pavel Chebotarev

Abstract: Let $m_{ij}$ be the mean first passage time from state $i$ to state $j$ in an $n$-state ergodic homogeneous Markov chain with transition matrix $T$. Let $G$ be the weighted digraph without loops whose vertex set is the set of states of the Markov chain and arc weights are equal to the corresponding transition probabilities. We give a graph-theoretic interpretation to $m_{ij}$. Namely, we show that $m_{ij}=1/q_j$ if $i=j$ and $m_{ij}=f_{ij}/(\sigma q_j)$ if $i\ne j$, where $f_{ij}$ is the total weight of 2-tree spanning converging forests in $G$ that have one tree containing $i$ and the other tree converging to $j$, $q_j$ is the total weight of spanning trees converging to $j$, and $\sigma$ is the total weight of spanning converging trees in $G$.

http://arXiv.org/abs/math/0701359
http://front.math.ucdavis.edu/math.PR/0701359 (alternate)

5121. Efficient estimation of the cardinality of large data sets

Author(s): Philippe Chassaing (IECN) and Lucas Gerin (IECN)

Abstract: F.Giroire has recently proposed an algorithm which returns the approximate number of distincts elements in a large sequence of words, under strong constraints coming from the analysis of large data bases. His estimation is based on statistical properties of uniform random variables in $[0,1]$. In this note we propose an optimal estimation, using Kullback information and estimation theory.

http://arXiv.org/abs/math/0701347
http://front.math.ucdavis.edu/math.ST/0701347 (alternate)

5122. Jarzynski's Identity

Author(s): Evelina Shamarova

Abstract: Jarzynski's identity (non-equilibrium work theorem) relates the equilibrium free energy difference $\Dl F$ to the work $W$ carried out on a system during a non-equilibrium transformation. In physics literature, the identity is usually written in the form: $ e^{-\beta W} = e^{-\beta\Dl F}$, where the average is said to be taken over all trajectories in the phase space. The identity in this form has been derived in different ways and published by many authors. Since the identity contains the "average over trajectories", it is natural to interpret this average as the expectation relative to a probability measure on trajectories, while assuming that the system evolves stochastically. In the present work, Jarzynski's identity is formulated and proved mathematically rigorous. It is written in the form $\mathbb E[e^{-\beta W}] = e^{-\beta\Dl F}$, where $\mathbb E$ is the expectation relative to a probability measure on phase space paths. For this probability measure, some analytical assumptions under which Jarzynki's identity holds, are found. Keywords: Probability measures on phase space paths, integration over phase space paths, non-equilibrium statistical mechanics, rigorous consideration of Jarzynski's identity

http://arXiv.org/abs/math/0701360
http://front.math.ucdavis.edu/math.PR/0701360 (alternate)

5123. A particle system in interaction with a rapidly varying environment: Mean field limits and applications

Author(s): Charles Bordenave and David McDonald and Alexandre Proutiere

Abstract: We study an interacting particle system whose dynamics depends on an interacting random environment. As the number of particles grows large, the transition rate of the particles slows down (perhaps because they share a common resource of fixed capacity). The transition rate of a particle is determined by its state, by the empirical distribution of all the particles and by a rapidly varying environment. The transitions of the environment are determined by the empirical distribution of the particles. We prove the propagation of chaos on the path space of the particles and establish that the limiting trajectory of the empirical measure of the states of the particles satisfies a deterministic differential equation. This deterministic differential equation involves the time averages of the environment process. We apply our results to analyze the performance of communication networks where users access some resources using random distributed multi-access algorithms. For these networks, we show that the environment process corresponds to a process describing the number of clients in a certain loss network, which allows us provide simple and explicit expressions of the network performance.

http://arXiv.org/abs/math/0701363
http://front.math.ucdavis.edu/math.PR/0701363 (alternate)

5124. On uniqueness of maximal coupling for diffusion processes with a reflection

Author(s): Kazumasa Kuwada

Abstract: A maximal coupling of two diffusion processes makes two diffusion particles meet as early as possible. We study the uniqueness of maximal couplings under a sort of "reflection structure" which ensures the existence of such couplings. In this framework, the uniqueness in the class of Markovian couplings holds for the Brownian motion on a Riemannian manifold whereas it fails in more singular cases. We also prove that a Kendall-Cranston coupling is maximal under the reflection structure.

http://arXiv.org/abs/math/0701372
http://front.math.ucdavis.edu/math.PR/0701372 (alternate)

5125. The birthday problem and Markov chain Monte Carlo

Author(s): Itai Benjamini and Ben Morris

Abstract: We study the problem of generating a sample from the stationary distribution of a Markov chain, given a method to simulate the chain. We give an approximation algorithm for the case of a random walk on a regular graph with n vertices that runs in expected time O^*(\sqrt{n} x L^2-mixing time). This is close to the best possible, since \sqrt{n} is a lower bound on the worst-case expected running time of any algorithm.

http://arXiv.org/abs/math/0701390
http://front.math.ucdavis.edu/math.PR/0701390 (alternate)

5126. Multivariate regular variation of heavy-tailed Markov chains

Author(s): Johan Segers

Abstract: The upper extremes of a Markov chain with regulary varying stationary marginal distribution are known to exhibit under general conditions a multiplicative random walk structure called the tail chain. More generally, if the Markov chain is allowed to switch from positive to negative extremes or vice versa, the distribution of the tail chain increment may depend on the sign of the tail chain on the previous step. But even then, the forward and backward tail chain mutually determine each other through a kind of adjoint relation. As a consequence, the finite-dimensional distributions of the Markov chain are multivariate regularly varying in a way determined by the back-and-forth tail chain. An application of the theory yields the asymptotic distribution of the past and the future of the solution to a stochastic difference equation conditionally on the present value being large in absolute value.

http://arXiv.org/abs/math/0701411
http://front.math.ucdavis.edu/math.PR/0701411 (alternate)

5127. Hydrodynamics for a non-conservative Interacting Particle System

Author(s): Glauco Valle

Abstract: We study a simple one-dimensional model which is roughly based on the spread of rainfall on a volume already occupied by a incompressible fluid aiming to describe the microscopic evolution of the density of mass of the fluid in infinite volume under local regular increase of mass of the system and obtain the macroscopic behaviour through the hydrodynamic limit.

http://arXiv.org/abs/math/0701413
http://front.math.ucdavis.edu/math.PR/0701413 (alternate)

5128. A Lower Bound on the Disconnection Time of a Discrete Cylinder

Author(s): Amir Dembo and Alain-Sol Sznitman

Abstract: We study the asymptotic behavior for large N of the disconnection time T_N of simple random walk on a discrete cylinder with base a d-dimensional discrete torus of side-length N. When d is sufficiently large, we are able to substantially improve the lower bounds obtained by the authors in a previous article when d is bigger or equal to 2. We show here that the laws of N^(2d)/T_N are tight.

http://arXiv.org/abs/math/0701414
http://front.math.ucdavis.edu/math.PR/0701414 (alternate)

5129. A phase transition for competition interfaces

Author(s): Pablo A. Ferrari and James B. Martin and Leandro P. R. Pimentel

Abstract: We study the competition interface between two growing clusters in a growth model associated to last-passage percolation. When the initial unoccupied set is approximately a cone, we show that this interface has an asymptotic direction with probability 1. The behaviour of this direction depends on the angle theta of the cone: for theta greater or equal to 180, the direction is deterministic, while for theta smaller than 180, it is random, and its distribution can be given explicitly in certain cases. We also obtain partial results on the fluctuations of the interface around its asymptotic direction. The evolution of the competition interface in the growth model can be mapped onto the path of a second-class particle in the totally asymmetric simple exclusion process; from the existence of the limiting direction for the interface, we obtain a new and rather natural proof of the strong law of large numbers (with perhaps a random limit) for the position of the second-class particle at large times.

http://arXiv.org/abs/math/0701418
http://front.math.ucdavis.edu/math.PR/0701418 (alternate)

5130. Tail Asymptotics for Discrete Event Systems

Author(s): Marc Lelarge

Abstract: In the context of communication networks, the framework of stochastic event graphs allows a modeling of control mechanisms induced by the communication protocol and an analysis of its performances. We concentrate on the logarithmic tail asymptotics of the stationary response time for a class of networks that admit a representation as (max,plus)-linear systems in a random medium. We are able to derive analytic results when the distribution of the holding times are light-tailed. We show that the lack of independence may lead in dimension bigger than one to non-trivial effects in the asymptotics of the sojourn time. We also study in detail a simple queueing network with multipath routing.

http://arXiv.org/abs/math/0701420
http://front.math.ucdavis.edu/math.PR/0701420 (alternate)

5131. A fractional generalization of the Poisson processes

Author(s): Francesco Mainardi and Rudolf Gorenflo and Enrico Scalas

Abstract: It is our intention to provide via fractional calculus a generalization of the pure and compound Poisson processes, which are known to play a fundamental role in renewal theory, without and with reward, respectively. We first recall the basic renewal theory including its fundamental concepts like waiting time between events, the survival probability, the counting function. If the waiting time is exponentially distributed we have a Poisson process, which is Markovian. However, other waiting time distributions are also relevant in applications, in particular such ones with a fat tail caused by a power law decay of its density. In this context we analyze a non-Markovian renewal process with a waiting time distribution described by the Mittag-Leffler function. This distribution, containing the exponential as particular case, is shown to play a fundamental role in the infinite thinning procedure of a generic renewal process governed by a power asymptotic waiting time. We then consider the renewal theory with reward that implies a random walk subordinated to a renewal process.

http://arXiv.org/abs/math/0701454
http://front.math.ucdavis.edu/math.PR/0701454 (alternate)

5132. Renewal processes of Mittag-Leffler and Wright type

Author(s): Francesco Mainardi and Rudolf Gorenflo and Alessandro Vivoli

Abstract: After sketching the basic principles of renewal theory we discuss the classical Poisson process and offer two other processes, namely the renewal process of Mittag-Leffler type and the renewal process of Wright type, so named by us because special functions of Mittag-Leffler and of Wright type appear in the definition of the relevant waiting times. We compare these three processes with each other, furthermore consider corresponding renewal processes with reward and numerically their long-time behaviour.

http://arXiv.org/abs/math/0701455
http://front.math.ucdavis.edu/math.PR/0701455 (alternate)

5133. Optimal control of a large dam, taking into account the water costs

Author(s): Vyacheslav M. Abramov

Abstract: Consider a large dam model, which is characterized by an upper level $L^{upper}$ and lower level $L^{lower}$, and if in time $t$ the level of water $L_t$ is between these bounds, then the dam is said to be in a normal state. The value $L$ = $L^{upper}$ - $L^{lower}$ is assumed to be large. The passage of lower or upper bounds leads to damage, the cost per time unit of which is $J_1=j_1L$ and $J_2=j_2L$ correspondingly, where $j_1$ and $j_2$ are given constants. Let $c_{L_t}$ denote a water cost, depending on the level of water in time $t$, $L^{lower}L^{upper}\}$ and $q_i$=$\lim_{t\to\infty}\mathbf{P}\{L_t=i\}$ ($L^{lower}

http://arXiv.org/abs/math/0701458
http://front.math.ucdavis.edu/math.PR/0701458 (alternate)

5134. Multivariate normal approximation using exchangeable pairs

Author(s): Sourav Chatterjee and Elizabeth Meckes

Abstract: Since the introduction of Stein's method in the early 1970s, much research has been done in extending and strengthening it; however, there does not exist a version of Stein's original method of exchangeable pairs for multivariate normal approximation. The aim of this article is to fill this void. We present two abstract normal approximation theorems using exchangeable pairs in multivariate contexts, one for situations in which the underlying symmetries are discrete, and one for situations involving continuous symmetry groups. We provide several illustrative examples, including a multivariate version of Hoeffding's combinatorial central limit theorem and a treatment of projections of Haar measure on the orthogonal and unitary groups.

http://arXiv.org/abs/math/0701464
http://front.math.ucdavis.edu/math.PR/0701464 (alternate)

5135. On Classical Analogues of Free Entropy Dimension

Author(s): A. Guionnet and D. Shlyakhtenko

Abstract: We define a classical probability analogue of Voiculescu's free entropy dimension that we shall call the classical probability entropy dimension of a probability measure on $\mathbb{R}^n$. We show that the classical probability entropy dimension of a measure is related with diverse other notions of dimension. First, it can be viewed as a kind of fractal dimension. Second, if one extends Bochner's inequalities to a measure by requiring that microstates around this measure asymptotically satisfy the classical Bochner's inequalities, then we show that the classical probability entropy dimension controls the rate of increase of optimal constants in Bochner's inequality for a measure regularized by convolution with the Gaussian law as the regularization is removed. We introduce a free analogue of the Bochner inequality and study the related free entropy dimension quantity. We show that it is greater or equal to the non-microstates free entropy dimension.

http://arXiv.org/abs/math/0701465
http://front.math.ucdavis.edu/math.PR/0701465 (alternate)

5136. On the hardness of sampling independent sets beyond the tree threshold

Author(s): Elchanan Mossel and Dror Weitz and Nicholas Wormald

Abstract: We consider local Markov chain Monte-Carlo algorithms for sampling from the weighted distribution of independent sets with activity $\l$, where the weight of an independent set $I$ is $\l^{|I|}$. A recent result has established that Gibbs sampling is rapidly mixing in sampling the distribution for graphs of maximum degree $d$ and $\l<\l_c(d)$, where $\l_c(d)$ is the critical activity for uniqueness of the Gibbs measure (i.e., for decay of correlations with distance in the weighted distribution over independent sets) on the $d$-regular infinite tree. We show that for $d \geq 3$, $\l$ just above $\l_c(d)$ with high probability over $d$-regular bipartite graphs, any local Markov chain Monte-Carlo algorithm takes exponential time before getting close to the stationary distribution. Our results provide a rigorous justification for ``replica'' method heuristics. These heuristics were invented in theoretical physics and are used in order to derive predictions on Gibbs measures on random graphs in terms of Gibbs measures on trees. We conjecture that $\l_c$ is in fact the exact threshold for this computational problem, i.e., that for $\l>\l_c$ it is NP-hard to approximate the above weighted sum overindependent sets to within a factor polynomial in the size of the graph.

http://arXiv.org/abs/math/0701471
http://front.math.ucdavis.edu/math.PR/0701471 (alternate)

5137. The Evolution of the Mixing Rate

Author(s): Nikolaos Fountoulakis and Bruce Reed

Abstract: In this paper we present a study of the mixing time of a random walk on the largest component of a supercritical random graph, also known as the giant component. We identify local obstructions that slow down the random walk, when the average degree d is at most (ln n lnln n)^{1/2}, proving that the mixing time in this case is O((ln n/d)^2) asymptotically almost surely. As the average degree grows these become negligible and it is the diameter of the largest component that takes over, yielding mixing time O(ln n/ln d). We proved these results during the 2003-04 academic year. Similar results but for constant d were later proved independently by I. Benjamini, G. Kozma and N. Wormald.

http://arXiv.org/abs/math/0701474
http://front.math.ucdavis.edu/math.CO/0701474 (alternate)

5138. The ghosts of the Ecole Normale. Life, death and destiny of Ren\'{e} Gateaux

Author(s): Laurent Mazliak (PMA and IMJ)

Abstract: The present paper deals with the life and some aspects of the scientific contribution of the mathematician Ren\'{e} Gateaux, killed during World War 1 at the age of 25. Though he died very young, he left interesting results in functional analysis. In particular, he was among the first to try to construct an integral over an infinite dimensional space. His ideas were extensively developed later by L\'{e}vy. Among other things, he interpreted Gateaux's integral in a probabilistic framework that later led to the construction of Wiener measure. This article tries to explain this singular personal and professional destiny in pre and postwar France. It also recalls the slaughter inflicted on French students during the conflict.

http://arXiv.org/abs/math/0701490
http://front.math.ucdavis.edu/math.HO/0701490 (alternate)

5139. Computation tree and strong spatial mixing in multi-spin systems

Author(s): Chandra Nair and Prasad Tetali

Abstract: This paper deals with the construction of a computation tree (hypertree) for interacting systems modeled using graphs (hypergraphs) that preserve the marginal probability of any vertex of interest. Local message passing equations have been used for some time to approximate the marginal probabilities in graphs but it is known that these equations are incorrect for graphs with loops. In this paper we construct, for any finite graph and a fixed vertex, a finite computation tree with appropriately defined boundary conditions so that the marginal probability on the tree at the vertex matches that on the graph. For several interacting systems, we show using our approach that if there is strong spatial mixing on an infinite regular tree, then one has strong spatial mixing for any given graph with maximum degree bounded by that of the regular tree. Thus we identify the regular tree as the worst case graph for the notion of strong spatial mixing.

http://arXiv.org/abs/math/0701494
http://front.math.ucdavis.edu/math.PR/0701494 (alternate)

5140. On singular integral and martingale transforms

Author(s): S. Geiss and S. Montgomery-Smith and E. Saksman

Abstract: Linear equivalences of norms of vector-valued singular integral operators and vector-valued martingale transforms are studied. In particular, it is shown that the UMD(p)-constant of a Banach space X equals the norm of the real (or the imaginary) part of the Beurling-Ahlfors singular integral operator, acting on the X-valued L^p-space on the plane. Moreover, replacing equality by a linear equivalence, this is found to be the typical property of even multipliers. A corresponding result for odd multipliers and the Hilbert transform is given.

http://arXiv.org/abs/math/0701516
http://front.math.ucdavis.edu/math.CA/0701516 (alternate)

5141. Penalizations of the Brownian motion by a functional of its local times

Author(s): Joseph Najnudel

Abstract: In this article, we study the family of probability measures (indexed by a positive real number t), obtained by penalization of the Brownian motion by a given functional of its local times at time t. We prove that this family tends to a limit measure when t goes to infinity if the functional satisfies some conditions of domination, and we check these conditions in several particular cases.

http://arXiv.org/abs/math/0701526
http://front.math.ucdavis.edu/math.PR/0701526 (alternate)

5142. Voter models with heterozygosity selection

Author(s): Anja Sturm and Jan Swart

Abstract: This paper studies variations of the usual voter model that favour types that are locally less common. Such models are dual to certain systems of branching annihilating random walks that are parity preserving. For both the voter models and their dual branching annihilating systems we determine all homogeneous invariant laws, and we study convergence to these laws started from other initial laws.

http://arXiv.org/abs/math/0701555
http://front.math.ucdavis.edu/math.PR/0701555 (alternate)

5143. Harmonic analysis on a finite homogeneous space

Author(s): Fabio Scarabotti and Filippo Tolli

Abstract: In this paper, we study harmonic analysis on finite homogeneous spaces whose associated permutation representation decomposes with multiplicity. After a careful look at Frobenius reciprocity and transitivity of induction, and the introduction of three types of spherical functions, we develop a theory of Gelfand Tsetlin bases for permutation representations. Then we study several concrete examples on the symmetric groups, generalizing the Gelfand pair of the Johnson scheme; we also consider statistical and probabilistic applications. After that, we consider the composition of two permutation representations, giving a non commutative generalization of the Gelfand pair associated to the ultrametric space; actually, we study the more general notion of crested product. Finally, we consider the exponentiation action, generalizing the decomposition of the Gelfand pair of the Hamming scheme; actually, we study a more general construction that we call wreath product of permutation representations, suggested by the study of finite lamplighter random walks. We give several examples of concrete decompositions of permutation representations and several explicit 'rules' of decomposition.

http://arXiv.org/abs/math/0701533
http://front.math.ucdavis.edu/math.RT/0701533 (alternate)

5144. Networks of Recurrent Events, a Theory of Records, and an Application to Finding Causal Signatures in Seismicity

Author(s): J. Davidsen and P. Grassberger and M. Paczuski

Abstract: We propose a method to search for signs of causal structure in spatiotemporal data making minimal a priori assumptions about the underlying dynamics. To this end, we generalize the elementary concept of recurrence for a point process in time to recurrent events in space and time. An event is defined to be a recurrence of any previous event if it is closer to it in space than all the intervening events. As such, each sequence of recurrences for a given event is a record breaking process. This definition provides a strictly data driven technique to search for structure. Defining events to be nodes, and linking each event to its recurrences, generates a network of recurrent events. Significant deviations in properties of that network compared to networks arising from random processes allows one to infer attributes of the causal dynamics that generate observable correlations in the patterns. We derive analytically a number of properties for the network of recurrent events composed by a random process. We extend the theory of records to treat not only the variable where records happen, but also time as continuous. In this way, we construct a fully symmetric theory of records leading to a number of new results. Those analytic results are compared to the properties of a network synthesized from earthquakes in Southern California. Significant disparities from the ensemble of acausal networks that can be plausibly attributed to the causal structure of seismicity are: (1) Invariance of network statistics with the time span of the events considered, (2) Appearance of a fundamental length scale for recurrences, independent of the time span of the catalog, which is consistent with observations of the ``rupture length'', (3) Hierarchy in the distances and times of subsequent recurrences.

http://arXiv.org/abs/physics/0701190
http://front.math.ucdavis.edu/physics/0701190 (alternate)

5145. Exit asymptotics for small diffusion about an unstable equilibrium

Author(s): Yuri Bakhtin

Abstract: A dynamical system perturbed by white noise in a neighborhood of an unstable fixed point is considered. We obtain the exit asymptotics in the limit of vanishing noise intensity. This is a refinement of a result by Kifer (1981).

http://arXiv.org/abs/math/0701569
http://front.math.ucdavis.edu/math.PR/0701569 (alternate)

5146. Generating Random Vectors in (Z/pZ)^d Via an Affine Random Process

Author(s): Martin Hildebrand and Joseph McCollum

Abstract: This paper considers some random processes of the form X_{n+1}=TX_n+B_n (mod p) where B_n and X_n are random variables over (Z/pZ)^d and T is a fixed d x d integer matrix which is invertible over the complex numbers. For a particular distribution for B_n, this paper improves results of Asci to show that if T has no complex eigenvalues of length 1, then for integers p relatively prime to det(T), order (log p)^2 steps suffice to make X_n close to uniformly distributed where X_0 is the zero vector. This paper also shows that if T has a complex eigenvalue which is a root of unity, then order p^b steps are needed for X_n to get close to uniform where b is a value which may depend on T and X_0 is the zero vector.

http://arXiv.org/abs/math/0701570
http://front.math.ucdavis.edu/math.PR/0701570 (alternate)

5147. Meinardus' theorem on weighted partitions: extensions and a probabilistic proof

Author(s): Boris L. Granovsky and Dudley Stark and Michael Erlihson

Abstract: We give a probalistic proof of the famous Meinardus' asymptotic formula for the number of weighted partitions with weakened one of the three Meinardus' conditions, and extend the resulting version of the theorem to other two classis types of decomposable combinatorial structures, which are called assemblies and selections. The results obtained are based on combining Meinardus' analytical approach with probabilistic method of Khitchine.

http://arXiv.org/abs/math/0701584
http://front.math.ucdavis.edu/math.PR/0701584 (alternate)

5148. Harmonic analysis of finite lamplighter random walks

Author(s): Fabio Scarabotti and Filippo Tolli

Abstract: Recently, several papers have been devoted to the analysis of lamplighter random walks, in particular when the underlying graph is the infinite path $\mathbb{Z}$. In the present paper, we develop a spectral analysis for lamplighter random walks on finite graphs. In the general case, we use the $C_2$-symmetry to reduce the spectral computations to a series of eigenvalue problems on the underlying graph. In the case the graph has a transitive isometry group $G$, we also describe the spectral analysis in terms of the representation theory of the wreath product $C_2\wr G$. We apply our theory to the lamplighter random walks on the complete graph and on the discrete circle. These examples were already studied by Haggstrom and Jonasson by probabilistic methods.

http://arXiv.org/abs/math/0701603
http://front.math.ucdavis.edu/math.PR/0701603 (alternate)

5149. Connected allocation to Poisson points in R^2

Author(s): Maxim Krikun (IECN)

Abstract: his note answers one question in [math.PR/0505668], concerning the connected allocation for the Poisson process in R^2. The proposed solution makes use of the Riemann map from the plane minus the minimal spanning forest of the Poisson point process to the halfplane. A picture of a numerically simulated example is included.

http://arXiv.org/abs/math/0701611
http://front.math.ucdavis.edu/math.PR/0701611 (alternate)

5150. Asymptotic enumeration of dense 0-1 matrices with specified line sums and forbidden positions

Author(s): Catherine Greenhill and Brendan D. McKay

Abstract: Let S=(s_1,s_2,..., s_m) and T = (t_1,t_2,..., t_n) be vectors of non-negative integers with \sum_{i=1}^m s_i = \sum_{j=1}^n t_j, and let X=(x_{jk}) be an m*n matrix over {0,1}. Define B(S,T,X) to be the number of m*n matrices B=(b_{jk}) over {0,1} with row sums given by S and column sums given by T such that x_{jk}=1 implies b_{jk}=0 for all j,k. That is, X specifies a set of entries of B required to be 0. Equivalently, B(S,T,X) is the number of bipartite graphs with m vertices in one part with degrees given by S, and n vertices in the other part with degrees given by T, and avoiding all the edges specified in X. Note that B(S,T,X)/B(S,T,0) is the probability that a uniformly chosen {0,1}-matrix with row sums S and column sums T has zeros in the places where X is nonzero. An asymptotic formula for B(S,T,X) was given by McKay (1984) in the case that the matrices are sparse. In the case of dense matrices there seem to be no prior results except for the special case X=0 studied by Canfield, Greenhill and McKay (math.CO/0606496). This paper extends the analytic methods used by the latter paper to obtain an asymptotic formula for B(S,T,X) in the dense regime where the entries of S and T can vary within certain limits and the row and column sums of X are not too large. As applications, we find the asymptotic number of simple digraphs with given vectors of in-degree and out-degree, and the expected permanent of a {0,1}-matrix with given row and column sums, with both results holding in the dense regime.

http://arXiv.org/abs/math/0701600
http://front.math.ucdavis.edu/math.CO/0701600 (alternate)

5151. A limit theorem for diffusions on graphs with variable configuration

Author(s): Alexey M. Kulik

Abstract: A limit theorem for a sequence of diffusion processes on graphs is proved in a case when vary both parameters of the processes (the drift and diffusion coefficients on every edge and the asymmetry coefficients in every vertex), and configuration of graphs, where the processes are set on. The explicit formulae for the parameters of asymmetry for the vertices of the limiting graph are given in the case, when, in the pre-limiting graphs, some groups of vertices form knots contracting into a points.

http://arXiv.org/abs/math/0701632
http://front.math.ucdavis.edu/math.PR/0701632 (alternate)

5152. Growth of preferential attachment random graphs via continuous-time branching processes

Author(s): K.B. Athreya and A.P. Ghosh and S. Sethuraman

Abstract: A version of ``preferential attachment'' random graphs, corresponding to linear ``weights'' with random ``edge additions,'' which generalizes some previously considered models, is studied. This graph model is embedded in a continuous-time branching scheme and, using the branching process apparatus, several results on the graph model asymptotics are obtained, some extending previous results, such as growth rates for a typical degree and the maximal degree, behavior of the vertex where the maximal degree is attained, and a law of large numbers for the empirical distribution of degrees which shows certain ``scale-free'' or ``power-law'' behaviors.

http://arXiv.org/abs/math/0701649
http://front.math.ucdavis.edu/math.PR/0701649 (alternate)

5153. The lower tail problem for homogeneous functionals of stable processes with no negative jumps

Author(s): Thomas Simon (DP)

Abstract: Let Z be a strictly a-stable real Levy process (a>1) and X be a fluctuating b-homogeneous additive functional of Z. We investigate the asymptotics of the first passage-time of X above 1, and give a general upper bound. When Z has no negative jumps, we prove that this bound is optimal and does not depend on the homogeneity parameter b. This extends a result of Y. Isozaki.

http://arXiv.org/abs/math/0701653
http://front.math.ucdavis.edu/math.PR/0701653 (alternate)

5154. Splitting pairs and the number of clusters generated by random pair incompatibilities

Author(s): Damien Pitman

Abstract: We consider a random fitness landscape on the space of haploid diallelic genotypes with n genetic loci, where each genotype is considered either inviable or viable depending on whether or not there are any incompatibilities among its allele pairs. We suppose that each allele pair in the set of all possible allele pairs on the n loci is independently incompatible with probability p=c/(2n). We examine the connectivity of the viable genotypes under single locus mutations and show that, for 01, there are no viable genotypes with probability converging to one. The genotype space is equivalent to the n-dimensional hypercube and the viable genotypes are solutions to a random 2-SAT problem, so the same result holds for the connectivity of solutions in the hypercube to a random 2-SAT problem.

http://arXiv.org/abs/math/0701656
http://front.math.ucdavis.edu/math.PR/0701656 (alternate)

5155. Normal form transforms separate slow and fast modes in stochastic dynamical systems

Author(s): A. J. Roberts

Abstract: Modelling stochastic systems has many important applications. Normal form coordinate transforms are a powerful way to untangle interesting long term macroscale dynamics from detailed microscale dynamics. We explore such coordinate transforms of stochastic differential systems when the dynamics has both slow modes and quickly decaying modes. The thrust is to derive normal forms useful for macroscopic modelling of complex stochastic microscopic systems. Thus we not only must reduce the dimensionality of the dynamics, but also endeavour to separate all slow processes from all fast time processes, both deterministic and stochastic. Quadratic stochastic effects in the fast modes contribute to the drift of the important slow modes. The results will help us accurately model, interpret and simulate multiscale stochastic systems.

http://arXiv.org/abs/math/0701623
http://front.math.ucdavis.edu/math.DS/0701623 (alternate)

5156. Asymptotic evolution of acyclic random mappings

Author(s): Steven N. Evans and Tye Lidman

Abstract: An acyclic mapping from an $n$ element set into itself is a mapping $\phi$ such that if $\phi^k(x) = x$ for some $k$ and $x$, then $\phi(x) = x$. Equivalently, $\phi^\ell = \phi^{\ell+1} = ...$ for $\ell$ sufficiently large. We investigate the behavior as $n \to \infty$ of a Markov chain on the collection of such mappings. At each step of the chain, a point in the $n$ element set is chosen uniformly at random and the current mapping is modified by replacing the current image of that point by a new one chosen independently and uniformly at random, conditional on the resulting mapping being again acyclic. We can represent an acyclic mapping as a directed graph (such a graph will be a collection of rooted trees) and think of these directed graphs as metric spaces with some extra structure. Heuristic calculations indicate that the metric space valued process associated with the Markov chain should, after an appropriate time and ``space'' rescaling, converge as $n \to \infty$ to a real tree ($\R$-tree) valued Markov process that is reversible with respect to a measure induced naturally by the standard reflected Brownian bridge. The limit process, which we construct using Dirichlet form methods, is a Hunt process with respect to a suitable Gromov-Hausdorff-like metric. This process is similar to one that appears in earlier work by Evans and Winter as the limit of chains involving the subtree prune and regraft tree (SPR) rearrangements from phylogenetics.

http://arXiv.org/abs/math/0701657
http://front.math.ucdavis.edu/math.PR/0701657 (alternate)

5157. Diffusive variance for a tagged particle in $d\leq 2$ asymmetric simple exclusion

Author(s): Sunder Sethuraman

Abstract: We study the equilibrium fluctuations of a tagged particle in finite-range simple exclusion processes on Z^d with biased single particle jump rates. It is known the variance of the tagged particle at time t is diffusive, that is on order O(t), in d\geq 3, and in d=1 when in addition the jump rate is nearest-neighbor, and moreover, in these cases, central limit theorems in diffusive scale have been proved. In this article, we give some partial results in the open cases in d\leq 2. Namely, we show diffusivity of the tagged particle variance at time t in the sense of some upper and lower bounds on order O(t) in d=2, and also in d=1 when in addition the jump rate is not nearest-neighbor. Also, a characterization of the tagged particle variance is given. The main methods are in analyzing H_{-1} norm variational inequalities.

http://arXiv.org/abs/math/0701660
http://front.math.ucdavis.edu/math.PR/0701660 (alternate)

5158. Critical Age Dependent Branching Markov Processes and Their Scaling Limits

Author(s): Krishna Athreya and Siva Athreya and and Srikanth Iyer

Abstract: This paper studies: (i) the long time behaviour of the empirical distribution of age and normalised position of an age dependent critical branching Markov process conditioned on non-extinction; and (ii) the super-process limit of a sequence of age dependent critical branching Brownian motions.

http://arXiv.org/abs/math/0701661
http://front.math.ucdavis.edu/math.PR/0701661 (alternate)

5159. Shape curvatures and transversal fluctuations in the first passage percolation model

Author(s): Yu Zhang

Abstract: We consider the first passage percolation model on the square lattice. In this model, $\{t(e): e{an edge of}{\bf Z}^2 \}$ is an independent identically distributed family with a common distribution $F$. We denote by $T({\bf 0}, v)$ the passage time from the origin to $v$ for $v\in {\bf R}^2$ and $B(t)=\{v\in {\bf R}^d: T({\bf 0}, v)\leq t\}.$ It is well known that if $F(0) < p_c$, there exists a compact shape ${\bf B}_F\subset {\bf R}^2$ such that for all $\epsilon >0$, $t {\bf B}_F(1-\epsilon) \subset {B(t)} \subset t{\bf B}_F(1+\epsilon)$, eventually with a probability 1. For each shape boundary point $u$, we denote its right- and left-curvature exponents by $\kappa^+(u)$ and $\kappa^-(u)$. In addition, for each vector $u$, we denote the transversal fluctuation exponent by $\xi(u)$. In this paper, we can show that $\xi(u) \leq 1-\max\{\kappa^-(u)/2, \kappa^+(u)/2\}$ for all shape boundary points $u$. To pursue a curvature on ${\bf B}_F$, we consider passage times with a special distribution infsupp$(F)=l$ and $F(l)=p > \vec{p}_c$, where $l$ is a positive number and $\vec{p}_c$ is a critical point for the oriented percolation model. With this distribution, it is known that there is a flat segment on the shape boundary between angles $0< \theta_p^- < \theta_p^+< 90^\circ$. In this paper, we show that the shape are strictly convex at the directions $\theta_p^\pm$. Moreover, we also show that for all $r>0$, $\xi((r, \theta^\pm_p)) = 0.5$ and $\xi((r, \theta)) =1$ for all $\theta_p^- <\theta< \theta_p^+$ and $r>0$.

http://arXiv.org/abs/math/0701689
http://front.math.ucdavis.edu/math.PR/0701689 (alternate)

5160. Spatial Epidemics: Critical Behavior in One Dimension

Author(s): Steven P. Lalley

Abstract: In the simple mean-field SIS and SIR epidemic models, infection is transmitted from infectious to susceptible members of a finite population by independent p-coin tosses. Spatial variants of these models are proposed, in which finite populations of size N are situated at the sites of a lattice and infectious contacts are limited to individuals at neighboring sites. Scaling laws for these models are given when the infection parameter p is such that the epidemics are critical. It is shown that in all cases there is a critical threshold for the numbers initially infected: below the threshold, the epidemic evolves in essentially the same manner as its branching envelope, but at the threshold evolves like a branching process with a size-dependent drift. The corresponding scaling limits are super-Brownian motions and Dawson-Watanabe processes with killing, respectively.

http://arXiv.org/abs/math/0701698
http://front.math.ucdavis.edu/math.PR/0701698 (alternate)

5161. Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws

Author(s): Alexander Gnedin and Ben Hansen and Jim Pitman

Abstract: This paper collects facts about the number of occupied boxes in the classical balls-in-boxes occupancy scheme with infinitely many positive frequencies: equivalently, about the number of species represented in samples from populations with infinitely many species. We present moments of this random variable, discuss asymptotic relations among them and with related random variables, and draw connections with regular variation, which appears in various manifestations.

http://arXiv.org/abs/math/0701718
http://front.math.ucdavis.edu/math.PR/0701718 (alternate)

5162. Distance estimates for dependent thinnings of point processes with densities

Author(s): Dominic Schuhmacher

Abstract: In [Schuhmacher, Electron. J. Probab. 10 (2005), 165--201] estimates of the Barbour-Brown distance d_2 between the distribution of a thinned point process and the distribution of a Poisson process were derived by combining discretization with a result based on Stein's method. In the present article we concentrate on point processes that have a density with respect to a Poisson process. For such processes we can apply a corresponding result directly without the detour of discretization and thus obtain better and more natural bounds not only in d_2 but also in the stronger total variation metric. We give applications for thinning by covering with an independent Boolean model and "Mat{\'e}rn type I"-thinning of fairly general point processes. These applications give new insight into the respective models, and either generalize or improve earlier results.

http://arXiv.org/abs/math/0701728
http://front.math.ucdavis.edu/math.PR/0701728 (alternate)

5163. Non-equilibrium stochastic dynamics in continuum: The free case

Author(s): Y. Kondratiev and E. Lytvynov and M. R\"ockner

Abstract: We study the problem of identification of a proper state-space for the stochastic dynamics of free particles in continuum, with their possible birth and death. In this dynamics, the motion of each separate particle is described by a fixed Markov process $M$ on a Riemannian manifold $X$. The main problem arising here is a possible collapse of the system, in the sense that, though the initial configuration of particles is locally finite, there could exist a compact set in $X$ such that, with probability one, infinitely many particles will arrive at this set at some time $t>0$. We assume that $X$ has infinite volume and, for each $\alpha\ge1$, we consider the set $\Theta_\alpha$ of all infinite configurations in $X$ for which the number of particles in a compact set is bounded by a constant times the $\alpha$-th power of the volume of the set. We find quite general conditions on the process $M$ which guarantee that the corresponding infinite particle process can start at each configuration from $\Theta_\alpha$, will never leave $\Theta_\alpha$, and has cadlag (or, even, continuous) sample paths in the vague topology. We consider the following examples of applications of our results: Brownian motion on the configuration space, free Glauber dynamics on the configuration space (or a birth-and-death process in $X$), and free Kawasaki dynamics on the configuration space. We also show that if $X=\mathbb R^d$, then for a wide class of starting distributions, the (non-equilibrium) free Glauber dynamics is a scaling limit of (non-equilibrium) free Kawasaki dynamics.

http://arXiv.org/abs/math/0701736
http://front.math.ucdavis.edu/math.PR/0701736 (alternate)

5164. Search cost for a nearly optimal path in a binary tree

Author(s): Robin Pemantle

Abstract: Consider a binary tree, to the vertices of which are assigned independent Bernoulli random variables with mean p <= 1/2. How many of these Bernoullis one must look at in order to find a path of length n from the root which maximizes, up to a factor of 1 - epsilon, the sum of the Bernoullis along the path? In the case, p = 1/2 (the critical value for nontriviality), it is shown to take of order epsilon^{-1} n steps. In the case p < 1/2, the number of steps is shown to be exponential in epsilon^{-1/2}. This last result matches Aldous' upper bound for a certain family of subcases.

http://arXiv.org/abs/math/0701741
http://front.math.ucdavis.edu/math.PR/0701741 (alternate)

5165. Exponential ergodicity of the solutions to SDE's with a jump noise

Author(s): Alexey M.Kulik

Abstract: The mild sufficient conditions for exponential ergodicity of a Markov process, defined as the solution to SDE with a jump noise, are given. These conditions include three principal claims: recurrence condition R, topological irreducibility condition S and non-degeneracy condition N, the latter formulated in the terms of a certain random subspace of \Re^m, associated with the initial equation. The examples are given, showing that, in general, none of three principal claims can be removed without losing ergodicity of the process. The key point in the approach, developed in the paper, is that the local Doeblin condition can be derived from N and S via the stratification method and criterium for the convergence in variations of the family of induced measures on \Re^m.

http://arXiv.org/abs/math/0701747
http://front.math.ucdavis.edu/math.PR/0701747 (alternate)

5166. Decay rates large deviations for the planar voter model occupation time

Author(s): G. Maillard and T. Mountford

Abstract: We study the decay rate of large deviation probabilities of occupation times, up to time $t$, for the voter model $\eta\colon\Z^2\times[0,\infty)\ra\{0,1\}$ with simple random walk transition kernel, starting from a Bernoulli product distribution with density $\rho\in(0,1)$. In \cite{bramcoxgri88}, Bramson, Cox and Griffeath showed that the decay rate order lies in $[\log(t),\log^2(t)]$. In this paper, we establish the true decay rates depending on the level. We show that the decay rates are $\log^2(t)$ when the deviation from $\rho$ is maximal (i.e., $\eta\equiv 0$ or 1), and $\log(t)$ in all other situations. This answers some conjecture in \cite{bramcoxgri88} and confirms analysis carried out in \cite{benfrakra96}, \cite{dorgod98} and \cite{howgod98}.

http://arXiv.org/abs/math/0701754
http://front.math.ucdavis.edu/math.PR/0701754 (alternate)

5167. Area distribution and scaling function for punctured polygons

Author(s): Christoph Richard and Iwan Jensen and Anthony J. Guttmann

Abstract: Punctured polygons are polygons with internal holes which are also polygons. The external and internal polygons are of the same type, and they are mutually as well as self-avoiding. We rigorously analyse the effect of a finite number of punctures on the limiting area distribution in a uniform ensemble, where punctured polygons with equal perimeter have the same probability of occurrence. The results rely on an assumption on the limiting area distribution for unpunctured polygons. Our analysis leads to conjectures about the possible scaling behaviour of the models. We also analyse exact enumeration data. For staircase polygons with punctures of fixed size, we find exact generating functions for the first few area-moments. For staircase polygons with punctures of arbitrary size, a careful numerical analysis yields very accurate estimates for the area-moments. Interestingly, we find that the leading correction term for each area-moment is proportional to the corresponding area-moment with one less puncture. We finally analyse corresponding quantities for punctured self-avoiding polygons and find agreement with the exact formulas to at least 3--4 significant digits.

http://arXiv.org/abs/math/0701633
http://front.math.ucdavis.edu/math.CO/0701633 (alternate)

5168. Parametrized Stochastic Grammars for RNA Secondary Structure Prediction

Author(s): Robert S. Maier

Abstract: We propose a two-level stochastic context-free grammar (SCFG) architecture for parametrized stochastic modeling of a family of RNA sequences, including their secondary structure. A stochastic model of this type can be used for maximum a posteriori estimation of the secondary structure of any new sequence in the family. The proposed SCFG architecture models RNA subsequences comprising paired bases as stochastically weighted Dyck-language words, i.e., as weighted balanced-parenthesis expressions. The length of each run of unpaired bases, forming a loop or a bulge, is taken to have a phase-type distribution: that of the hitting time in a finite-state Markov chain. Without loss of generality, each such Markov chain can be taken to have a bounded complexity. The scheme yields an overall family SCFG with a manageable number of parameters.

http://arXiv.org/abs/q-bio/0701036
http://front.math.ucdavis.edu/q-bio.BM/0701036 (alternate)

5169. Learning Trigonometric Polynomials from Random Samples and Exponential Inequalities for Eigenvalues of Random Matrices

Author(s): Karlheinz Groechenig and Benedikt M. Poetscher and Holger Rauhut

Abstract: Motivated by problems arising in random sampling of trigonometric polynomials, we derive exponential inequalities for the operator norm of the difference between the sample second moment matrix $n^{-1}U^*U$ and its expectation where $U$ is a complex random $n\times D$ matrix with independent rows. These results immediately imply deviation inequalities for the largest (smallest) eigenvalues of the sample second moment matrix, which in turn lead to results on the condition number of the sample second moment matrix. We also show that trigonometric polynomials in several variables can be learned from $const \cdot D \ln D$ random samples.

http://arXiv.org/abs/math/0701781
http://front.math.ucdavis.edu/math.PR/0701781 (alternate)

5170. Occupation laws for some time-nonhomogeneous Markov chains

Author(s): Zach Dietz and Sunder Sethuraman

Abstract: We consider finite-state time-nonhomogeneous Markov chains where the probability of moving from state $i$ to state $j\neq i$ at time $n$ is $G(i,j)/n^\zeta$ for a ``generator'' matrix $G$ and strength parameter $\zeta>0$. In these chains, as time grows, the positions are less and less likely to change, and so form simple models of age-dependent time-reinforcing behaviors. These chains, however, exhibit some different, perhaps unexpected, asymptotic occupation laws depending on parameters. Although on the one hand it is shown that the asymptotic position converges to a point-mixture for all $\zeta>0$, on the other hand, the average position, when variously $0<\zeta<1$, $\zeta>1$ or $\zeta=1$, is shown to converges to a constant, a point-mixture, or a distribution $\mu_G$ with no atoms and full support on a certain simplex respectively. The last type of limit can be seen as a sort of ``spreading'' between the cases $0<\zeta<1$ and $\zeta>1$. In particular, when $G$ is appropriately chosen, $\mu_G$ is a Dirichlet distribution with certain parameters, reminiscent of results in Polya urns.

http://arXiv.org/abs/math/0701798
http://front.math.ucdavis.edu/math.PR/0701798 (alternate)

5171. Weak convergence of step processes and an application for critical multitype branching processes with immigration

Author(s): M\'arton Isp\'any and Gyula Pap

Abstract: First, sufficient conditions are given for a system $(U^n_k)_{n\in\NN, k\in\ZZ_+}$ of random variables in $\RR^d$ and for a diffusion process $(\cU_t)_{t\in\RR_+}$ such that $\cU^n\distr\cU$, where $\cU^n_t:=\sum_{k=0}^{\nt}U^n_k$. Next, sufficient conditions are given for a system $(\psi_{n,k})_{n\in\NN, k\in\ZZ_+}$ of Borel functions $\psi_{n,k}:(\RR^d)^{k+1}\to\RR^p$ and for a measurable mapping $\Psi:\DD(\RR^d)\to\DD(\RR^p)$ such that $(\cU^n,\cV^n,\cY^n)\distr(\cU,\cV,\cY)$, where $\cV^n_t:=V^n_{\nt}$ with $V^n_k:=\psi_{n,k}(U^n_0,...,U^n_k)$, $\cV:=\Psi(\cU)$, $\cY^n_t:=\sum_{k=1}^{\nt}V^n_{k-1}\otimes U^n_k$, and $\cY_t:=\int_0^t\cV_s\otimes\dd\cU_s$. As an application of these results, first a Feller type diffusion approximation is derived for critical multitype branching processes with immigration if the offspring mean matrix is primitive, then the asymptotic behavior of the conditional least squares estimator of the offspring mean matrix is established.

http://arXiv.org/abs/math/0701803
http://front.math.ucdavis.edu/math.PR/0701803 (alternate)

5172. The cutoff phenomenon for randomized riffle shuffles

Author(s): Guan-Yu Chen and Laurent Saloff-Coste

Abstract: We study the cutoff phenomenon for generalized riffle shuffles where, at each step, the deck of cards is cut into a random number of packs of multinomial sizes which are then riffled together.

http://arXiv.org/abs/math/0701827
http://front.math.ucdavis.edu/math.PR/0701827 (alternate)

5173. M/M/$\infty$ queues in quasi-Markovian Random Environment

Author(s): B. D'Auria

Abstract: In this paper we investigate an M/M/$\infty$ queue whose parameters depend on an external random environment that we assume to be a quasi-Markovian process with finite state space. For this model we show a recursive formula that allows to compute all the factorial moments for the number of customers in the system in steady state. The used technique is based on the calculation of the row moments of the area of a bidimensional random set. Finally some examples where it is possible to get explicit formulas are given together with comparisons with previous known results.

http://arXiv.org/abs/math/0701842
http://front.math.ucdavis.edu/math.PR/0701842 (alternate)

5174. BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces

Author(s): Philippe Briand (IRMAR) and Fulvia Confortola

Abstract: This paper is devoted to the study of the differentiability of solutions to real-valued backward stochastic differential equations (BSDEs for short) with quadratic generators driven by a cylindrical Wiener process. The main novelty of this problem consists in the fact that the gradient equation of a quadratic BSDE has generators which satisfy stochastic Lipschitz conditions involving BMO martingales. We show some applications to the nonlinear Kolmogorov equations.

http://arXiv.org/abs/math/0701849
http://front.math.ucdavis.edu/math.PR/0701849 (alternate)

5175. Extremal Probabilistic Problems and Hotelling's T^2 Test Under Symmetry Condition

Author(s): Iosif Pinelis

Abstract: We consider Hotelling's T^2 statistic for an arbitrary d-dimensional sample. If the sampling is not too deterministic or inhomogeneous, then under zero means hypothesis, T^2 tends to \chi^2_d in distribution. We show that a test for the orthant symmetry condition introduced by Efron can be constructed which does not essentially differ from the one based on \chi^2_d and at the same time is applicable not only for large random homogeneous samples but for all multidimensional samples without exceptions. The main assertions have the form of inequalities, not that of limit theorems; these inequalities are exact representing the solutions to certain extremal problems. Let us also mention an auxiliary result which itself may be of interest: \chi_d-(d-1)^{1/2} decreases in distribution in d to its limit N(0,1/2).

http://arXiv.org/abs/math/0701806
http://front.math.ucdavis.edu/math.ST/0701806 (alternate)

5176. Rate of convergence of penalized-likelihood context tree estimators

Author(s): Florencia G. Leonardi

Abstract: We find upper bounds for the probability of error of the penalized-likelihood type context tree estimators, where the trees are not assumed to be finite. This estimators includes the well-known Bayesian Information Criterion (BIC). We show that the maximal decay for the probability of error can be achieved with a penalized term of the form $n^\alpha$, with $0 < \alpha < 1$.

http://arXiv.org/abs/math/0701810
http://front.math.ucdavis.edu/math.ST/0701810 (alternate)

5177. Deterministic modal Bayesian Logic: derive the Bayesian inference within the modal logic T

Author(s): Frederic Dambreville (DGA/CTA/DT/GIP)

Abstract: In this paper a conditional logic is defined and studied. This conditional logic, DmBL, is constructed as a deterministic counterpart to the Bayesian conditional. The logic is unrestricted, so that any logical operations are allowed. A notion of logical independence is also defined within the logic itself. This logic is shown to be non-trivial and is not reduced to classical propositions. A model is constructed for the logic. Completeness results are proved. It is shown that any unconditioned probability can be extended to the whole logic DmBL. The Bayesian conditional is then recovered from the probabilistic DmBL. At last, it is shown why DmBL is compliant with Lewis' triviality.

http://arXiv.org/abs/math/0701801
http://front.math.ucdavis.edu/math.LO/0701801 (alternate)

5178. Free diffusions and Matrix models with strictly convex interaction

Author(s): A. Guionnet and D. Shlyakhtenko

Abstract: We study solutions to the free stochastic differential equation $dX_t = dS_t - \half DV(X_t)dt$, where $V$ is a locally convex polynomial potential in $m$ non-commuting variables. We show that for self-adjoint $V$, the law $\mu_V$ of a stationary solution is the limit law of a random matrix model, in which an $m$-tuple of self-adjoint matrices are chosen according to the law $\exp(-N \textrm{Tr}(V(A_1,...,A_m)))dA_1... dA_m$. We show that if $V=V_\beta$ depends on complex parameters $\beta_1,...,\beta_k$, then the law $\mu_V$ is analytic in $\beta$ at least for those $\beta$ for which $V_\beta$ is locally convex. In particular, this gives information on the region of convergence of the generating function for planar maps. We show that the solution $dX_t$ has nice convergence properties with respect to the operator norm. This allows us to derive several properties of $C^*$ and $W^*$ algebras generated by an $m$-tuple with law $\mu_V$. Among them is lack of projections, exactness, the Haagerup property, and embeddability into the ultrapower of the hyperfinite II$_1$ factor. We show that the microstates free entropy $\chi(\tau_V)$ is finite. A corollary of these results is the fact that the support of the law of any self-adjoint polynomial in $X_1,...,X_n$ under the law $\mu_V$ is connected, vastly generalizing the case of a single random matrix.

http://arXiv.org/abs/math/0701787
http://front.math.ucdavis.edu/math.OA/0701787 (alternate)

5179. Classical and Variational Differentiability of BSDEs with quadratic growth

Author(s): Stefan Ankirchner and Peter Imkeller and Goncalo Reis

Abstract: We consider Backward Stochastic Differential Equations (BSDE) with generators that grow quadratically in the control variable. In a more abstract setting, we first allow both the terminal condition and the generator to depend on a vector parameter $x$. We give sufficient conditions for the solution pair of the BSDE to be differentiable in $x$. These results can be applied to systems of forward-backward SDE. If the terminal condition of the BSDE is given by a sufficiently smooth function of the terminal value of a forward SDE, then its solution pair is differentiable with respect to the initial vector of the forward equation. Finally we prove sufficient conditions for solutions of quadratic BSDE to be differentiable in the variational sense (Malliavin differentiable).

http://arXiv.org/abs/math/0701875
http://front.math.ucdavis.edu/math.PR/0701875 (alternate)

5180. Propagation of chaos and Poincar\'{e} inequalities for a system of particles interacting through their cdf

Author(s): Benjamin Jourdain (CERMICS) and Florent Malrieu (IRMAR)

Abstract: In the particular case of a concave flux function, we are interested in the long time behaviour of the nonlinear process associated to the one-dimensional viscous scalar conservation law. We also consider the particle system obtained by remplacing the cumulative distribution function in the drift coefficient of this nonlinear process by the empirical cdf. We first obtain trajectorial propagation of chaos result. Then, Poincar\'{e} inequalities are used to get explicit estimates concerning the long time behaviour of both the nonlinear process and the particle system.

http://arXiv.org/abs/math/0701879
http://front.math.ucdavis.edu/math.PR/0701879 (alternate)

5181. Local Gaussian fluctuations in the Airy and discrete PNG processes

Author(s): Jonas H\"agg

Abstract: We prove that the Airy process, A(t), locally fluctuates like a Brownian motion. In the same spirit we also show that in a certain scaling limit, the so called discrete polynuclear growth (PNG) process behaves like a Brownian motion.

http://arXiv.org/abs/math/0701880
http://front.math.ucdavis.edu/math.PR/0701880 (alternate)

5182. Ricci curvature of Markov chains on metric spaces

Author(s): Yann Ollivier

Abstract: We define the Ricci curvature of Markov chains on metric spaces as a local contraction coefficient of the random walk acting on the space of probability measures equipped with a Wasserstein transportation distance. For Brownian motion on a Riemannian manifold this gives back the value of Ricci curvature of a tangent vector. Examples of positively curved spaces for this definition include the discrete cube and discrete versions of the Ornstein--Uhlenbeck process. Moreover this generalization is consistent with the Bakry--\'Emery Ricci curvature for Brownian motion with a drift on a Riemannian manifold. Positive Ricci curvature is easily shown to imply a spectral gap and a L\'evy--Gromov-like Gaussian concentration theorem. These bounds are sharp in several interesting examples.

http://arXiv.org/abs/math/0701886
http://front.math.ucdavis.edu/math.PR/0701886 (alternate)

5183. Measure-preserving transformations of Volterra Gaussian processes and related bridges

Author(s): Celine Jost

Abstract: We consider Volterra Gaussian processes on [0,T], where T>0 is a fixed time horizon. These are processes of type X_t=\int^t_0 z_X(t,s)dW_s, t\in[0,T], where z_X is a square-integrable kernel, and W is a standard Brownian motion. An example is fractional Brownian motion. By using classical techniques from operator theory, we derive measure-preserving transformations of X, and their inherently related bridges of X. As a closely connected result, we obtain a Fourier-Laguerre series expansion for the first Wiener chaos of a Gaussian martingale over [0,\infty).

http://arXiv.org/abs/math/0701888
http://front.math.ucdavis.edu/math.PR/0701888 (alternate)

5184. Expansion properties of a random regular graph after random vertex deletions

Author(s): Catherine Greenhill (University of New South Wales) and Fred B. Holt (University of Washington), Nicholas Wormald (University of Waterloo)

Abstract: We investigate the following vertex percolation process. Starting with a random regular graph of constant degree, delete each vertex independently with probability p, where p=n^{-alpha} and alpha=alpha(n) is bounded away from 0. We show that a.a.s. the resulting graph has a connected component of size n-o(n) which is an expander, and all other components are trees of bounded size. Sharper results are obtained with extra conditions on alpha. These results have an application to the cost of repairing a certain peer-to-peer network after random failures of nodes.

http://arXiv.org/abs/math/0701863
http://front.math.ucdavis.edu/math.CO/0701863 (alternate)

5185. Record indices and age-ordered frequencies in Gibbs random partitions

Author(s): Robert C. Griffiths and Dario Span\'{o}

Abstract: The distribution of age-ordered frequencies arising from an exchangeable Gibbs partition is studied in relation with the distribution of the positions at which new mutations appear in a sample.

http://arXiv.org/abs/math/0701897
http://front.math.ucdavis.edu/math.PR/0701897 (alternate)

5186. Differentiating sigma-fields for Gaussian and shifted Gaussian processes

Author(s): S\'{e}bastien Darses (PMA) and Ivan Nourdin (PMA) and Giovanni Peccati (LSTA)

Abstract: We study the notions of differentiating and non-differentiating sigma-fields in the general framework of (possibly drifted) Gaussian processes, and characterize their invariance properties under equivalent changes of probability measure. As an application, we investigate the class of stochastic derivatives associated with shifted fractional Brownian motions. We finally establish conditions for the existence of a jointly measurable version of the differentiated process, and we outline a general framework for stochastic embedded equations.

http://arXiv.org/abs/math/0701910
http://front.math.ucdavis.edu/math.PR/0701910 (alternate)

5187. Local limit theorems for ladder epochs

Author(s): Vladimir Vatutin and Vitali Wachtel

Abstract: Let {S_n, n=0,1,2,...} be a random walk generated by a sequence of i.i.d. random variables X_1, X_2,... and let tau be the first descending ladder epoch. Assuming that the distribution of X_1 belongs to the domain of attraction of an alpha-stable law, we study the asymptotic behavior of P(tau=n).

http://arXiv.org/abs/math/0701914
http://front.math.ucdavis.edu/math.PR/0701914 (alternate)

5188. Proliferating parasites in dividing cells : Kimmel's branching model revisited

Author(s): Vincent Bansaye (PMA)

Abstract: We consider a branching model introduced by M. Kimmel for cell division with parasite infection. Cells contain proliferating parasites which are shared randomly between the two daughter cells when they divide. We determine the probability that the organism recovers, meaning that the asymptotic proprotion of contaminated cells vanishes. We study the tree of contaminated cells, give the asymptotic number of contaminated cells and the asymptotic proportions of contaminated cells with a given number of parasites. This depends on domains inherited from the behavior of branching processes in random environment (BPRE) and given by the bivariate value of the means of parasite offsprings. In one of these domains, the convergence of proportions holds in probability, the limit is deterministic and given by the Yaglom quasistationary distribution. Moreover we get an interpretation of the limit of the Q-process as the size-biased quasistationary distribution.

http://arXiv.org/abs/math/0701917
http://front.math.ucdavis.edu/math.PR/0701917 (alternate)

5189. On lower limits and equivalences for distribution tails of randomly stopped sums

Author(s): Denis Denisov and Serguei Foss and Dmitry Korshunov

Abstract: For a distribution $F^{*\tau}$ of a random sum $S_\tau=\xi_1+...+\xi_\tau$ of i.i.d. random variables with a common distribution $F$ on the half-line $[0,\infty)$, we study the limits of the ratios of tails $\bar{F^{*\tau}}(x)/\bar F(x)$ as $x\to\infty$ (here $\tau$ is an independent counting random variable). We also consider applications of obtained results to random walks, compound Poisson distributions, infinitely divisible laws, and sub-critical branching processes.

http://arXiv.org/abs/math/0701920
http://front.math.ucdavis.edu/math.PR/0701920 (alternate)

5190. On several two-boundary problems for a particular class of L\'{e}vy processes

Author(s): Tetyana Kadankova and No\"{e}l Veraverbeke

Abstract: Several two-boundary problems are solved for a special L\'{e}vy process: the Poisson process with an exponential component. The jumps of this process are controlled by a homogeneous Poisson process, the positive jump size distribution is arbitrary, while the distribution of the negative jumps is exponential. Closed form expressions are obtained for the integral transforms of the joint distribution of the first exit time from an interval and the value of the overshoot through boundaries at the first exit time. Also the joint distribution of the first entry time into the interval and the value of the process at this time instant are determined in terms of integral transforms.

http://arXiv.org/abs/math/0701924
http://front.math.ucdavis.edu/math.PR/0701924 (alternate)

5191. Kernel Methods in Machine Learning

Author(s): Thomas Hofmann and Bernhard Sch\"olkopf and Alexander J. Smola

Abstract: We review machine learning methods employing positive definite kernels. These methods formulate learning and estimation problems in a reproducing kernel Hilbert space (RKHS) of functions defined on the data domain, expanded in terms of a kernel. Working in linear spaces of function has the benefit of facilitating the construction and analysis of learning algorithms while at the same time allowing large classes of functions. The latter include nonlinear functions as well as functions defined on non-vectorial data. We cover a wide range of methods, ranging from binary classifiers to sophisticated methods for estimation with structured data.

http://arXiv.org/abs/math/0701907
http://front.math.ucdavis.edu/math.ST/0701907 (alternate)

5192. Evaluation of Formal Posterior Distributions via Markov Chain Arguments

Author(s): Morris L. Eaton and James P. Hobert and Galin L. Jones and Wen-Lin Lai

Abstract: We consider evaluation of proper posterior distributions obtained from improper prior distributions. Our context is estimating a bounded function $\phi$ of a parameter when the loss is quadratic. If the posterior mean of $\phi$ is admissible for all bounded $\phi$ the posterior is \textit{strongly admissible}. In this paper, we present sufficient conditions for strong admissibility. These conditions involve the recurrence of a symmetric Markov chain associated with the estimation problem. We develop general sufficient conditions for recurrence of general state space Markov chains that are also of independent interest. Our main example concerns the $p$-dimensional multivariate normal distribution with mean vector $\theta$ when the prior distribution has the form $g_{0}(\theta) d\theta$ on the parameter space $\mathbb{R}^{p}$. Conditions on $g_{0}$ for strong admissibility of the posterior are provided.

http://arXiv.org/abs/math/0701938
http://front.math.ucdavis.edu/math.ST/0701938 (alternate)

5193. Exponential control of overlap in the replica method for p-spin Sherrington-Kirkpatrick model

Author(s): Dmitry Panchenko

Abstract: Recently, Michel Talagrand computed the large deviations limit $\lim_{N\to\infty}(Na)^{-1}\log \e Z_N^a$ for the moments of the partition function $Z_N$ in the Sherrington-Kirkpatrick model for all real $a\geq 0.$ For $a\geq 1$ the limit is given by Guerra's inverse bound and this result extends the classical physicist's replica method that corresponds to integer $a.$ We give a new proof for $a\geq 1$ in the case of the pure $p$-spin SK model that provides a strong exponential control of the overlap.

http://arXiv.org/abs/math-ph/0701074
http://front.math.ucdavis.edu/math-ph/0701074 (alternate)

5194. Asymptotics of non-intersecting Brownian motions and a 4 x 4 Riemann-Hilbert problem

Author(s): Evi Daems and Arno Kuijlaars and and Wim Veys

Abstract: We consider n one-dimensional Brownian motions, such that n/2 Brownian motions start at time t=0 in the starting point a and end at time t=1 in the endpoint b and the other n/2 Brownian motions start at time t=0 at the point -a and end at time t=1 in the point -b, conditioned that the n Brownian paths do not intersect in the whole time interval (0,1). The correlation functions of the positions of the non-intersecting Brownian motions have a determinantal form with a kernel that is expressed in terms of multiple Hermite polynomials of mixed type. We analyze this kernel in the large n limit for the case ab<1/2. We find that the limiting mean density of the positions of the Brownian motions is supported on one or two intervals and that the correlation kernel has the usual scaling limits from random matrix theory, namely the sine kernel in the bulk and the Airy kernel near the edges.

http://arXiv.org/abs/math/0701923
http://front.math.ucdavis.edu/math.CV/0701923 (alternate)

5195. A combinatorial method for calculating the moments of L\'evy area

Author(s): Daniel Levin and Mark Wildon

Abstract: We present a new way to compute the moments of the L\'evy area of a two-dimensional Brownian motion. Our approach uses iterated integrals and combinatorial arguments involving the shuffle product.

http://arXiv.org/abs/math/0702002
http://front.math.ucdavis.edu/math.PR/0702002 (alternate)

5196. Finite size scaling for the core of large random hypergraphs

Author(s): Amir Dembo and Andrea Montanari

Abstract: The (two) core of an hyper-graph is the maximal collection of hyper-edges within which no vertex appears only once. It is of importance in tasks such as efficiently solving a large linear system over GF[2], or iterative decoding of low-density parity-check codes used over the binary erasure channel. Similar structures emerge in a variety of NP-hard combinatorial optimization and decision problems, from vertex cover to satisfiability. For a uniformly chosen random hyper-graph of m=n\rho vertices and n hyper-edges, each consisting of the same fixed number l >= 3 of vertices, the size of the core exhibits for large n a first order phase transition, changing from o(n) for rho> rho_c to a positive fraction of n for rho0. Analyzing the corresponding `leaf removal' algorithm, we determine the associated finite size scaling behavior. In particular, if rho is inside the scaling window (more precisely, rho = rho_c+r n^{-1/2}, the probability of having a core of size Theta(n) has a limit strictly between 0 and 1, and a leading correction of order Theta(n^{-1/6}). The correction admits a sharp characterization in terms of the distribution of a Brownian motion with quadratic shift, from which it inherits the scaling with n. This behavior is expected to be universal for wide collection of combinatorial problems.

http://arXiv.org/abs/math/0702007
http://front.math.ucdavis.edu/math.PR/0702007 (alternate)

5197. On Stein's method and perturbations

Author(s): Andrew D. Barbour and Vydas Cekanavicius and Aihua Xia

Abstract: Stein's (1972) method is a very general tool for assessing the quality of approximation of the distribution of a random element by another, often simpler, distribution. In applications of Stein's method, one needs to establish a Stein identity for the approximating distribution, solve the Stein equation and estimate the behaviour of the solutions in terms of the metrics under study. For some Stein equations, solutions with good properties are known; for others, this is not the case. Barbour and Xia (1999) introduced a perturbation method for Poisson approximation, in which Stein identities for a large class of compound Poisson and translated Poisson distributions are viewed as perturbations of a Poisson distribution. In this paper, it is shown that the method can be extended to very general settings, including perturbations of normal, Poisson, compound Poisson, binomial and Poisson process approximations in terms of various metrics such as the Kolmogorov, Wasserstein and total variation metrics. Examples are provided to illustrate how the general perturbation method can be applied.

http://arXiv.org/abs/math/0702008
http://front.math.ucdavis.edu/math.PR/0702008 (alternate)

5198. On a randomized PNG model with a columnar defect

Author(s): Vincent Beffara (UMPA-ENSL) and Vladas Sidoravicius (BR-IMPA) and Maria Eulalia Vares (BR-CBPF)

Abstract: We study a variant of poly-nuclear growth where the level boundaries perform continuous-time, discrete-space random walks, and study how its asymptotic behavior is affected by the presence of a columnar defect on the line. We prove that there is a non-trivial phase transition in the strength of the perturbation, above which the law of large numbers for the height function is modified.

http://arXiv.org/abs/math/0702012
http://front.math.ucdavis.edu/math.PR/0702012 (alternate)

5199. A functional CLT for the occupation time of state-dependent branching random walk

Author(s): Matthias Birkner and Iljana Z\"ahle

Abstract: We show that the centred occupation time process of the origin of a system of critical binary branching random walks in dimension $d \ge 3$, started off either from a Poisson field or in equilibrium, when suitably normalised, converges to a Brownian motion in $d \ge 4$. In $d=3$, the limit process is fractional Brownian motion with Hurst parameter 3/4 when starting in equilibrium, and a related Gaussian process when starting from a Poisson field. For (dependent) branching random walks with state dependent branching rate we obtain convergence in f.d.d. to the same limit process, and for $d=3$ also a functional limit theorem.

http://arXiv.org/abs/math/0702020
http://front.math.ucdavis.edu/math.PR/0702020 (alternate)

5200. Graphs with specified degree distributions, simple epidemics and local vaccination strategies

Author(s): Tom Britton and Svante Janson and Anders Martin-Lof

Abstract: Consider a random graph, having a pre-specified degree distribution F but other than that being uniformly distributed, describing the social structure (friendship) in a large community. Suppose one individual in the community is externally infected by an infectious disease and that the disease has its course by assuming that infected individuals infect their not yet infected friends independently with probability p. For this situation the paper determines R_0 and tau_0, the basic reproduction number and the asymptotic final size in case of a major outbreak. Further, the paper looks at some different local vaccination strategies where individuals are chosen randomly and vaccinated, or friends of the selected individuals are vaccinated, prior to the introduction of the disease. For the studied vaccination strategies the paper determines R_v: the reproduction number, and tau_v: the asymptotic final proportion infected in case of a major outbreak, after vaccinating a fraction v.

http://arXiv.org/abs/math/0702021
http://front.math.ucdavis.edu/math.PR/0702021 (alternate)

5201. Wigner random matrices with non-symmetrically distributed entries

Author(s): Sandrine Peche and Alexander Soshnikov

Abstract: We show that the spectral radius of an $N\times N$ random symmetric matrix with i.i.d. bounded centered but non-symmetrically distributed entries is bounded from above by $ 2 \*\sigma + o(N^{-6/11+\epsilon}), $ where $\sigma^2 $ is the variance of the matrix entries and $\epsilon $ is an arbitrary small positive number. Our bound improves the earlier results by Z.F\"{u}redi and J.Koml\'{o}s (1981), and the recent bound obtained by Van Vu (2005).

http://arXiv.org/abs/math/0702035
http://front.math.ucdavis.edu/math.PR/0702035 (alternate)

5202. On the Variance of the Optimal Alignment Score for an Asymmetric Scoring Function

Author(s): Christian Houdr\'e and Heinrich Matzinger

Abstract: We investigate the variance of the optimal alignment score of two independent iid binary, with parameter 1/2, sequences of length $n$. The scoring function is such that one letter has a somewhat larger score than the other letter. In this setting, we prove that the variance is of order $n$, and this confirms Waterman's conjecture in this case.

http://arXiv.org/abs/math/0702036
http://front.math.ucdavis.edu/math.PR/0702036 (alternate)

5203. A large deviation principle in H\"older norm for multiple fractional integrals

Author(s): Marta Sanz-Sol\'e and Iv\'an Torrecilla-Tarantino

Abstract: For a fractional Brownian motion $B^H$ with Hurst parameter $H\in]{1/4},{1/2}[\cup]{1/2},1[$, multiple indefinite integrals on a simplex are constructed and the regularity of their sample paths are studied. Then, it is proved that the family of probability laws of the processes obtained by replacing $B^H$ by $\epsilon^{{1/2}} B^H$ satisfies a large deviation principle in H\"older norm. The definition of the multiple integrals relies upon a representation of the fractional Brownian motion in terms of a stochastic integral with respect to a standard Brownian motion. For the large deviation principle, the abstract general setting given by Ledoux in [Lecture Notes in Math., vol. 1426 (1990) 1-14] is used.

http://arXiv.org/abs/math/0702049
http://front.math.ucdavis.edu/math.PR/0702049 (alternate)

5204. H\"{o}lder regularity for operator scaling stable random fields

Author(s): Hermine Bierm\'{e} (MAP5) and C\'{e}line Lacaux (IECN)

Abstract: We investigate the sample paths regularity of operator scaling alpha-stable random fields. Such fields were introduced as anisotropic generalizations of self-similar fields and satisfy a scaling property for a real matrix E. In the case of harmonizable operator scaling random fields, the sample paths are locally H\"{o}lderian and their H\"{o}lder regularity is characterized by the eigen decomposition with respect to E. In particular, the directional H\"{o}lder regularity may vary and is given by the eigenvalues of E. In the case of moving average operator scaling random alpha-stable random fields, with 0

http://arXiv.org/abs/math/0702050
http://front.math.ucdavis.edu/math.PR/0702050 (alternate)

5205. Large deviations for empirical path measures in cycles of integer partitions

Author(s): Stefan Adams

Abstract: Consider a large system of $N$ Brownian motions in $\mathbb{R}^d$ on some fixed time interval $[0,\beta]$ with symmetrised initial-terminal condition. That is, for any $i$, the terminal location of the $i$-th motion is affixed to the initial point of the $\sigma(i)$-th motion, where $\sigma$ is a uniformly distributed random permutation of $1,...,N$. In this paper, we describe the large-N behaviour of the empirical path measure (the mean of the Dirac measures in the $N$ paths) when $ \Lambda\uparrow\mathbb{R}^d $ and $ N/|\Lambda|\to\rho $. The rate function is given as a variational formula involving a certain entropy functional and a Fenchel-Legendre transform. Depending on the dimension and the density $ \rho $, there is phase transition behaviour for the empirical path measure. For certain parameters (high density, large time horizon) and dimensions $ d\ge 3 $ the empirical path measure is not supported on all paths $ [0,\infty)\to\mathbb{R}^d $ which contain a bridge path of any finite multiple of the time horizon $ [0,\beta] $. For dimensions $ d=1,2 $, and for small densities and small time horizon $ [0,\beta] $ in dimensions $ d\ge 3$, the empirical path measure is supported on those paths. In the first regime a finite fraction of the motions lives in cycles of infinite length. We outline that this transition leads to an empirical path measure interpretation of {\it Bose-Einstein condensation}, known for systems of Bosons.

http://arXiv.org/abs/math/0702053
http://front.math.ucdavis.edu/math.PR/0702053 (alternate)

5206. Mixtures in non stable Levy processes

Author(s): Nicola Cufaro Petroni

Abstract: We analyze the Levy processes produced by means of two interconnected classes of non stable, infinitely divisible distribution: the Variance Gamma and the Student laws. While the Variance Gamma family is closed under convolution, the Student one is not: this makes its time evolution more complicated. We prove that -- at least for one particular type of Student processes suggested by recent empirical results, and for integral times -- the distribution of the process is a mixture of other types of Student distributions, randomized by means of a new probability distribution. The mixture is such that along the time the asymptotic behavior of the probability density functions always coincide with that of the generating Student law. We put forward the conjecture that this can be a general feature of the Student processes. We finally analyze the Ornstein--Uhlenbeck process driven by our Levy noises and show a few simulation of it.

http://arXiv.org/abs/math/0702058
http://front.math.ucdavis.edu/math.PR/0702058 (alternate)

5207. Line-of-sight percolation

Author(s): Bela Bollobas and Svante Janson and Oliver Riordan

Abstract: Given $\omega\ge 1$, let $Z^2_{(\omega)}$ be the graph with vertex set $Z^2$ in which two vertices are joined if they agree in one coordinate and differ by at most $\omega$ in the other. (Thus $Z^2_{(1)}$ is precisely $Z^2$.) Let $p_c(\omega)$ be the critical probability for site percolation in $Z^2_{(\omega)}$. Extending recent results of Frieze, Kleinberg, Ravi and Debany, we show that $\lim_{\omega\to\infty} \omega\pc(\omega)=\log(3/2)$. We also prove analogues of this result on the $n$-by-$n$ grid and in higher dimensions, the latter involving interesting connections to Gilbert's continuum percolation model. To prove our results, we explore the component of the origin in a certain non-standard way, and show that this exploration is well approximated by a certain branching process.

http://arXiv.org/abs/math/0702061
http://front.math.ucdavis.edu/math.PR/0702061 (alternate)

5208. Ancestral processes with selection: Branching and Moran models

Author(s): E. Baake and R. Bialowons

Abstract: We consider two versions of stochastic population models with mutation and selection. The first approach relies on a multitype branching process; here, individuals reproduce and change type (i.e., mutate) independently of each other, without restriction on population size. We analyze the equilibrium behaviour of this model, both in the forward and in the backward direction of time; the backward point of view emerges if the ancestry of individuals chosen randomly from the present population is traced back into the past. The second approach is the Moran model with selection. Here, the population has constant size N. Individuals reproduce (at rates depending on their types), the offspring inherits the parent's type, and replaces a randomly chosen individual (to keep population size constant). Independently of the reproduction process, individuals can change type. As in the branching model, we consider the ancestral lines of single individuals chosen from the equilibrium population. We use analytical results of Fearnhead (2002) to determine the explicit properties, and parameter dependence, of the ancestral distribution of types, and its relationship with the stationary distribution in forward time.

http://arXiv.org/abs/q-bio/0702002
http://front.math.ucdavis.edu/q-bio.PE/0702002 (alternate)

5209. Isospin asymmetry in nuclei and nuclear symmetry energy

Author(s): Tapan Mukhopadhyay and D.N. Basu

Abstract: Binding energy of isospin asymmetric nuclei can be accessed with minimally modified formula along the lines of the liquid droplet model by partitioning the symmetry term into volume and surface terms. The volume symmetry energy coefficient extracted from finite nuclei provides a constraint on the nuclear symmetry energy. This approach also yields the neutron skin of a finite nucleus through its relationship with the volume and surface symmetry terms and the Coulomb energy coefficient. The symmetry energy at saturation density obtained from the isoscalar as well as isovector components of the density dependent M3Y effective interaction is found to be in close agreement with the volume symmetry energy coefficient extracted from the measured atomic masses.

http://arXiv.org/abs/nucl-th/0605001
http://front.math.ucdavis.edu/nucl-th/0605001 (alternate)

5210. Computing the Loewner driving process of random curves in the half plane

Author(s): Tom Kennedy

Abstract: We simulate several models of random curves in the half plane and numerically compute their stochastic driving process (as given by the Loewner equation). Our models include models whose scaling limit is the Schramm-Loewner evolution (SLE) and models for which it is not. We study several tests of whether the driving process is Brownian motion. We find that just testing the normality of the process at a fixed time is not effective at determining if the process is Brownian motion. Tests that involve the independence of the increments of Brownian motion are much more effective. We also study the zipper algorithm for numerically computing the driving function of a simple curve. We give an implementation of this algorithm which runs in a time O(N^1.35) rather than the usual O(N^2), where N is the number of points on the curve.

http://arXiv.org/abs/math/0702071
http://front.math.ucdavis.edu/math.PR/0702071 (alternate)

5211. Limit theorems on locally compact Abelian groups

Author(s): Matyas Barczy and Alexander Bendikov and Gyula Pap

Abstract: We prove limit theorems for row sums of a rowwise independent infinitesimal array of random variables with values in a locally compact Abelian group. First we give a proof of Gaiser's theorem, since it does not have an easy access and it is not complete. This theorem gives sufficient conditions for convergence of the row sums, but the limit measure can not have a nondegenerate idempotent factor. Then we prove necessary and sufficient conditions for convergence of the row sums, where the limit measure can be also a nondegenerate Haar measure on a compact subgroup. Finally, we investigate special cases: the torus group, the group of p-adic integers and the p-adic solenoid.

http://arXiv.org/abs/math/0702078
http://front.math.ucdavis.edu/math.PR/0702078 (alternate)

5212. On a non-classical invariance principle

Author(s): Youri Davydov (Universite de Lille 1) and Vladimir Rotar (San Diego State University)

Abstract: We consider the invariance principle without the classical condition of asymptotic negligibility of individual terms. More precisely, we explore the difference of the following two distributions in the space C (of continuous functions on [0,1]). The first is the distribution of the continuous piecewise linear partial-sum process generated by a sequence of independent random variables, and the second is the distribution of the similar process generated by the sequence of normal r.v.'s with the same first two moments. The novelty is that the condition of negligibility of the r.v.'s is not imposed. We establish a necessary and sufficient condition of the weak convergence of the difference mentioned to zero measure in C.

http://arXiv.org/abs/math/0702085
http://front.math.ucdavis.edu/math.PR/0702085 (alternate)

5213. Divergence theorems in path space III: hypoelliptic diffusions and beyond

Author(s): Denis Bell

Abstract: Let $x$ denote a diffusion process defined on a closed compact manifold. In an earlier article, the author introduced a new approach to constructing admissible vector fields on the associated space of paths, under the assumption of ellipticity of $x$. In this article, this method is extended to yield similar results for degenerate diffusion processes. In particular, these results apply to non-elliptic diffusions satisfying H\"ormander's condition.

http://arXiv.org/abs/math/0702092
http://front.math.ucdavis.edu/math.PR/0702092 (alternate)

5214. Extinction versus unbounded growth; Habilitation Thesis of the University Erlangen-N\"urnberg

Author(s): Jan M. Swart

Abstract: Certain Markov processes, or deterministic evolution equations, have the property that they are dual to a stochastic process that exhibits extinction versus unbounded growth, i.e., the total mass in such a process either becomes zero, or grows without bounds as time tends to infinity. If this is the case, then this phenomenon can often be used to determine the invariant measures, or fixed points, of the process originally under consideration, and to study convergence to equilibrium. This principle, which has been known since early work on multitype branching processes, is here demonstrated on three new examples with applications in the theory of interacting particle systems.

http://arXiv.org/abs/math/0702095
http://front.math.ucdavis.edu/math.PR/0702095 (alternate)

5215. A note on ergodic transformations of self-similar Volterra Gaussian processes

Author(s): Celine Jost

Abstract: We derive a class of ergodic transformations of self-similar Gaussian processes that are Volterra, i.e. of type X_t = int^t_0 z_X(t,s)dW_s, t>0, where z_X is a deterministic kernel and W is a standard Brownian motion.

http://arXiv.org/abs/math/0702096
http://front.math.ucdavis.edu/math.PR/0702096 (alternate)

5216. Random Walk in Markovian Enviroment

Author(s): Dmitry Dolgopyat and Gerhard Keller and and Carlangelo Liverani

Abstract: We prove a quenched central limit theorem for random walks with bounded increments in a randomly evolving environment on Zd. We assume that the transition probabilities of the walk depend not too strongly on the environment and that the evolution of the environment is Markovian with strong spatial and temporal mixing properties.

http://arXiv.org/abs/math/0702100
http://front.math.ucdavis.edu/math.PR/0702100 (alternate)

5217. Tail probabilities for infinite series of regularly varying random vectors

Author(s): Henrik Hult and Gennady Samorodnitsky

Abstract: A random vector $X$ with representation $X = \sum_{j \geq 0} A_j Z_j$ is considered. Here $(Z_j)$ is a sequence of independent and identically distributed random vectors and $(A_j)$ is a sequence of random matrices, ``predictable'' with respect to the sequence $(Z_j)$. The distribution of $Z_1$ is assumed to be multivariate regular varying. Moment conditions on the matrices $(A_j)$ are determined under which the distribution of $X$ is regularly varying and, in fact, ``inherits'' its regular variation from that of $(Z_j)$'s. We compute the associated limiting measure. Examples include linear processes, random coefficient linear processes such as stochastic recurrence equations, random sums, and stochastic integrals.

http://arXiv.org/abs/math/0702112
http://front.math.ucdavis.edu/math.PR/0702112 (alternate)

5218. Dynamical properties of a tagged particle in the totally asymmetric simple exclusion process with the step-type initial condition

Author(s): T. Imamura and T. Sasamoto

Abstract: The one-dimensional totally asymmetric simple exclusion process (TASEP) is considered. We study the time evolution property of a tagged particle in TASEP with the step-type initial condition. Calculated is the multi-time joint distribution function of its position. Using the relation of the dynamics of TASEP to the Schur process, we show that the function is represented as the Fredholm determinant. We also study the scaling limit. The universality of the largest eigenvalue in the random matrix theory is realized in the limit. When the hopping rates of all particles are the same, it is found that the joint distribution function converges to that of the Airy process after the time at whichthe particle begins to move. On the other hand, when there are several particles with small hopping rate in front of a tagged particle, the limiting process changes at a certain time from the Airy process to the process of the largest eigenvalue in the Hermitian multi-matrix model with external sources.

http://arXiv.org/abs/math-ph/0702009
http://front.math.ucdavis.edu/math-ph/0702009 (alternate)

5219. Mellin transform and subordination laws in fractional diffusion processes

Author(s): Francesco Mainardi and Gianni Pagnini and Rudolf Gorenflo

Abstract: The Mellin transform is usually applied in probability theory to the product of independent random variables. In recent times the machinery of the Mellin transform has been adopted to describe the L\'evy stable distributions, and more generally the probability distributions governed by generalized diffusion equations of fractional order in space and/or in time. In these cases the related stochastic processes are self-similar and are simply referred to as fractional diffusion processes. We provide some integral formulas involving the distributions of these processes that can be interpreted in terms of subordination laws.

http://arXiv.org/abs/math/0702133
http://front.math.ucdavis.edu/math.PR/0702133 (alternate)

5220. Local Energy Statistics in Directed Polymers

Author(s): Irina Kourkova (PMA)

Abstract: Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should, in most circumstances, be the same as in the random energy model. We show that this conjecture holds true as well for directed polymers in random environment. We also show that, under certain conditions, this conjecture holds for directed polymers even if energy levels that grow moderately with the volume of the system are considered.

http://arXiv.org/abs/math/0702149
http://front.math.ucdavis.edu/math.PR/0702149 (alternate)

5221. Statistical analysis of the Diffie-Hellman key exchange protocol in a finite group

Author(s): I. Florescu and A. Myasnikov and A. Mahalanobis

Abstract: This paper presents a novel methodology to test the security of the Diffie-Hellman public key exchange protocol. The security of many cryptographic schemes rely on the hardness of this problem. We are presenting a purely statistical test to compare this problem in different groups. We are using groups included in the Zp group with p prime as a major example, however the methods presented are not restricted to these groups. The presentation of the results is primarily intended to introduce novel applications of statistical methodologies to the area of mathematical cryptography. As such we will emphasize the cryptographical aspects of the work more than the statistical notions.

http://arXiv.org/abs/math/0702155
http://front.math.ucdavis.edu/math.ST/0702155 (alternate)

5222. Pathwise inequalities for local time: applications to Skorokhod embeddings and optimal stopping

Author(s): A.M.G.Cox and D.Hobson and J.Ob\l\'oj

Abstract: We develop a class of pathwise inequalities of the form $H(B_t)\ge M_t+ F(L_t)$, where $B_t$ is Brownian motion, $L_t$ its local time at zero and $M_t$ a local martingale. The concrete nature of the representation makes the inequality useful for a variety of applications. In this work, we use the inequalities to derive constructions and optimality results of Vallois' Skorokhod embeddings. We discuss their financial interpretation in the context of robust pricing and hedging of options written on the local time. In the final part of the paper we use the inequalities to solve a class of optimal stopping problems of the form $\sup_\tau E[F(L_\tau)-\int_0^\tau \beta(B_s)ds]$. The solution is given via a minimal solution to a system of differential equations and thus resembles the maximality principle described by Peskir. Throughout, the emphasis is placed on the novelty and simplicity of the techniques.

http://arXiv.org/abs/math/0702173
http://front.math.ucdavis.edu/math.PR/0702173 (alternate)

5223. Difference approximation for local times of multidimensional diffusions

Author(s): Alexey M. Kulik

Abstract: We consider sequences of additive functionals of difference approximations for uniformly non-degenerate multidimensional diffusions. The conditions are given, sufficient for such a sequence to converge weakly to a W-functional of the limiting process. The class of the W-functionals, that can be obtained as the limiting ones, is completely described in the terms of the associated W-measures, and coincides with the class of the functionals that are regular w.r.t. the phase variable.

http://arXiv.org/abs/math/0702175
http://front.math.ucdavis.edu/math.PR/0702175 (alternate)

5224. Diffusion approximation for equilibrium Kawasaki dynamics in continuum

Author(s): Y.G. Kondratiev and O.V. Kutoviy and E.W. Lytvynov

Abstract: A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in $\mathbb R^d$ which randomly hop over the space. In this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs measure $\mu$ as invariant measure. We study a diffusive limit of such a dynamics, derived through a scaling of both the jump rate and time. Under weak assumptions on the potential of pair interaction, $\phi$, (in particular, admitting a singularity of $\phi$ at zero), we prove that, on a set of smooth local functions, the generator of the scaled dynamics converges to the generator of an equilibrium diffusive dynamics of an infinite system of interacting particles. If the set on which the generators converge is a core for the diffusion generator, the latter result implies the weak convergence of finite-dimensional distributions of the corresponding equilibrium processes. In particular, if the potential $\phi$ is from $C_{\mathrm b}^3(\R^d)$ and sufficiently quickly converges to zero at infinity, we conclude from a result in [Choi {\it et al.}, {J. Math. Phys.} {39} (1998) 6509--6536] that the convergence of processes holds when the limiting diffusion is the gradient stochastic dynamics.

http://arXiv.org/abs/math/0702178
http://front.math.ucdavis.edu/math.PR/0702178 (alternate)

5225. Finite-size effects for anisotropic bootstrap percolation: logarithmic corrections

Author(s): Aernout C.D. van Enter and Tim Hulshof

Abstract: In this note we analyze an anisotropic, two-dimensional bootstrap percolation model introduced by Gravner and Griffeath. We present upper and lower bounds on the finite-size effects. We discuss the similarities with the semi-oriented model introduced by Duarte.

http://arXiv.org/abs/cond-mat/0702145
http://front.math.ucdavis.edu/cond-mat/0702145 (alternate)

5226. Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps

Author(s): M.T. Barlow and R.F. Bass and and T. Kumagai

Abstract: We investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the corresponding heat kernel estimates.

http://arXiv.org/abs/math/0702221
http://front.math.ucdavis.edu/math.PR/0702221 (alternate)

5227. Rigorous confidence intervals for critical probabilities

Author(s): Oliver Riordan and Mark Walters

Abstract: We use the method of Balister, Bollobas and Walters to give rigorous 99.9999% confidence intervals for the critical probabilities for site and bond percolation on the 11 Archimedean lattices. In our computer calculations, the emphasis is on simplicity and ease of verification, rather than obtaining the best possible results. Nevertheless, we obtain intervals of width at most 0.0005 in all cases.

http://arXiv.org/abs/math/0702232
http://front.math.ucdavis.edu/math.PR/0702232 (alternate)

5228. A randomized Kaczmarz algorithm with exponential convergence

Author(s): Thomas Strohmer and Roman Vershynin

Abstract: The Kaczmarz method for solving linear systems of equations is an iterative algorithm that has found many applications ranging from computer tomography to digital signal processing. Despite the popularity of this method, useful theoretical estimates for its rate of convergence are still scarce. We introduce a randomized version of the Kaczmarz method for consistent, overdetermined linear systems and we prove that it converges with expected exponential rate. Furthermore, this is the first solver whose rate does not depend on the number of equations in the system. The solver does not even need to know the whole system, but only a small random part of it. It thus outperforms all previously known methods on general extremely overdetermined systems. Even for moderately overdetermined systems, numerical simulations as well as theoretical analysis reveal that our algorithm can converge faster than the celebrated conjugate gradient algorithm. Furthermore, our theory and numerical simulations confirm a prediction of Feichtinger et al. in the context of reconstructing bandlimited functions from nonuniform sampling.

http://arXiv.org/abs/math/0702226
http://front.math.ucdavis.edu/math.NA/0702226 (alternate)

5229. A simple proof of Kaijser's unique ergodicity result for hidden Markov $\alpha$-chains

Author(s): Fred Kochman and Jim Reeds

Abstract: According to a 1975 result of T. Kaijser, if some nonvanishing product of hidden Markov model (HMM) stepping matrices is subrectangular, and the underlying chain is aperiodic, the corresponding $\alpha$-chain has a unique invariant limiting measure $\lambda$. Here the $\alpha$-chain $\{\alpha_n\}=\{(\alpha_{ni})\}$ is given by \[\alpha_{ni}=P(X_n=i| Y_n,Y_{n-1},...),\] where $\{(X_n,Y_n)\}$ is a finite state HMM with unobserved Markov chain component $\{X_n\}$ and observed output component $\{Y_n\}$. This defines $\{\alpha_n\}$ as a stochastic process taking values in the probability simplex. It is not hard to see that $\{\alpha_n\}$ is itself a Markov chain. The stepping matrices $M(y)=(M(y)_{ij})$ give the probability that $(X_n,Y_n)=(j,y)$, conditional on $X_{n-1}=i$. A matrix is said to be subrectangular if the locations of its nonzero entries forms a cartesian product of a set of row indices and a set of column indices. Kaijser's result is based on an application of the Furstenberg--Kesten theory to the random matrix products $M(Y_1)M(Y_2)... M(Y_n)$. In this paper we prove a slightly stronger form of Kaijser's theorem with a simpler argument, exploiting the theory of e chains.

http://arXiv.org/abs/math/0702248
http://front.math.ucdavis.edu/math.PR/0702248 (alternate)

5230. Continuous-time mean-variance efficiency: the 80% rule

Author(s): Xun Li and Xun Yu Zhou

Abstract: This paper studies a continuous-time market where an agent, having specified an investment horizon and a targeted terminal mean return, seeks to minimize the variance of the return. The optimal portfolio of such a problem is called mean-variance efficient \`{a} la Markowitz. It is shown that, when the market coefficients are deterministic functions of time, a mean-variance efficient portfolio realizes the (discounted) targeted return on or before the terminal date with a probability greater than 0.8072. This number is universal irrespective of the market parameters, the targeted return and the length of the investment horizon.

http://arXiv.org/abs/math/0702249
http://front.math.ucdavis.edu/math.PR/0702249 (alternate)

5231. Periodicity in the transient regime of exhaustive polling systems

Author(s): I. M. MacPhee and M. V. Menshikov and S. Popov and S. Volkov

Abstract: We consider an exhaustive polling system with three nodes in its transient regime under a switching rule of generalized greedy type. We show that, for the system with Poisson arrivals and service times with finite second moment, the sequence of nodes visited by the server is eventually periodic almost surely. To do this, we construct a dynamical system, the triangle process, which we show has eventually periodic trajectories for almost all sets of parameters and in this case we show that the stochastic trajectories follow the deterministic ones a.s. We also show there are infinitely many sets of parameters where the triangle process has aperiodic trajectories and in such cases trajectories of the stochastic model are aperiodic with positive probability.

http://arXiv.org/abs/math/0702252
http://front.math.ucdavis.edu/math.PR/0702252 (alternate)

5232. Sample path large deviations for multiclass feedforward queueing networks in critical loading

Author(s): Kurt Majewski

Abstract: We consider multiclass feedforward queueing networks with first in first out and priority service disciplines at the nodes, and class dependent deterministic routing between nodes. The random behavior of the network is constructed from cumulative arrival and service time processes which are assumed to satisfy an appropriate sample path large deviation principle. We establish logarithmic asymptotics of large deviations for waiting time, idle time, queue length, departure and sojourn-time processes in critical loading. This transfers similar results from Puhalskii about single class queueing networks with feedback to multiclass feedforward queueing networks, and complements diffusion approximation results from Peterson. An example with renewal inter arrival and service time processes yields the rate function of a reflected Brownian motion. The model directly captures stationary situations.

http://arXiv.org/abs/math/0702256
http://front.math.ucdavis.edu/math.PR/0702256 (alternate)

5233. On the Hausdorff dimension of regular points of inviscid Burgers equation with stable initial data

Author(s): Thomas Simon (DP)

Abstract: Consider an inviscid Burgers equation whose initial data is a Levy a-stable process Z with a > 1. We show that when Z has positive jumps, the Hausdorff dimension of the set of Lagrangian regular points associated with the equation is strictly smaller than 1/a, as soon as a is close to 1. This gives a negative answer to a conjecture of Janicki and Woyczynski. Along the way, we contradict a recent conjecture of Z. Shi about the lower tails of integrated stable processes.

http://arXiv.org/abs/math/0702260
http://front.math.ucdavis.edu/math.PR/0702260 (alternate)

5234. Stochastic Models for Phylogenetic Trees on Higher-order Taxa

Author(s): David Aldous and Maxim Krikun and and Lea Popovic

Abstract: Simple stochastic models for phylogenetic trees on species have been well studied. But much paleontology data concerns time series or trees on higher-order taxa, and any broad picture of relationships between extant groups requires use of higher-order taxa. A coherent model for trees on (say) genera should involve both a species-level model and a model for the classification scheme by which species are assigned to genera. We present a general framework for such models, and describe three alternate classification schemes. Combining with the species-level model of Aldous-Popovic (2005), one gets models for higher-order trees, and we initiate analytic study of such models. In particular we derive formulas for the lifetime of genera, for the distribution of number of species per genus, and for the offspring structure of the tree on genera.

http://arXiv.org/abs/q-bio/0702014
http://front.math.ucdavis.edu/q-bio.PE/0702014 (alternate)

5235. Homogenization of periodic linear degenerate PDEs

Author(s): Martin Hairer and Etienne Pardoux

Abstract: It is well-known under the name of `periodic homogenization' that, under a centering condition of the drift, a periodic diffusion process on R^d converges, under diffusive rescaling, to a d-dimensional Brownian motion. Existing proofs of this result all rely on uniform ellipticity or hypoellipticity assumptions on the diffusion. In this paper, we considerably weaken these assumptions in order to allow for the diffusion coefficient to even vanish on an open set. As a consequence, it is no longer the case that the effective diffusivity matrix is necessarily non-degenerate. It turns out that, provided that some very weak regularity conditions are met, the range of the effective diffusivity matrix can be read off the shape of the support of the invariant measure for the periodic diffusion. In particular, this gives some easily verifiable conditions for the effective diffusivity matrix to be of full rank. We also discuss the application of our results to the homogenization of a class of elliptic and parabolic PDEs.

http://arXiv.org/abs/math/0702304
http://front.math.ucdavis.edu/math.PR/0702304 (alternate)

5236. A quenched invariance principle for certain ballistic random walks in i.i.d. environments

Author(s): Noam Berger and Ofer Zeitouni

Abstract: We prove that every random walk in i.i.d. environment in dimension greater than or equal to 4 that has an almost sure positive speed in a certain direction, an annealed invariance principle and some mild integrability condition for regeneration times also satisfies a quenched invariance principle. The argument is based on intersection estimates and a theorem of Bolthausen and Sznitman.

http://arXiv.org/abs/math/0702306
http://front.math.ucdavis.edu/math.PR/0702306 (alternate)

5237. Existence and smoothness of the density for spatially homogeneous SPDEs

Author(s): David Nualart (University of Kansas) and Lluis Quer-Sardanyons (Universitat Autonoma de Barcelona)

Abstract: In this paper, we extend Walsh's stochastic integral with respect to a Gaussian noise, white in time and with some homogeneous spatial correlation, in order to be able to integrate some random measure-valued processes. This extension turns out to be equivalent to Dalang's one. Then we study existence and regularity of the density of the probability law for the real-valued mild solution to a general second order stochastic partial differential equation driven by such a noise. For this, we apply the techniques of the Malliavin calculus. Our results apply to the case of the stochastic heat equation in any space dimension and the stochastic wave equation in space dimension $d=1,2,3$. Moreover, for these particular examples, known results in the literature have been improved.

http://arXiv.org/abs/math/0702312
http://front.math.ucdavis.edu/math.PR/0702312 (alternate)

5238. Measures with zeros in the inverse of their moment matrix

Author(s): J. William Helton and Jean B. Lasserre and Mihai Putinar

Abstract: We investigate and discuss when the inverse of a multivariate truncated moment matrix of a measure $\mu$ has zeros in some prescribed entries. We describe precisely which pattern of these zeroes corresponds to independence, namely, the measure having a product structure. A more refined finding is that the key factor forcing a zero entry in this inverse matrix is a certain conditional triangularity property of the orthogonal polynomials associated with the measure $\mu$.

http://arXiv.org/abs/math/0702314
http://front.math.ucdavis.edu/math.PR/0702314 (alternate)

5239. Convergence of weighted power variations of fractional Brownian motion

Author(s): Mihai Gradinaru (IECN) and Ivan Nourdin (PMA)

Abstract: The first part of the paper contains the study of the convergence for some weighted power variations of a fractional Brownian motion B with Hurst index H in (0,1). The behaviour is different when H<1/2 and powers are odd, compared with the case when H=1/2 or when H>1/2 and powers are even. In the second part, one applies the results of the first part to compute the exact rate of convergence of some approximating schemes associated to scalar stochastic differential equations driven by B. The limit of the error between the exact solution and the considered scheme (whose size depends on the Hurst index H) is computed explicitly.

http://arXiv.org/abs/math/0702317
http://front.math.ucdavis.edu/math.PR/0702317 (alternate)

5240. Neighbor Selection and Hitting Probability in Small-World Graphs

Author(s): Oskar Sandberg

Abstract: Small-world graphs, which combine randomized and structured elements, are seen as prevalent in nature. Jon Kleinberg showed that in some graphs of this type it is possible to route, or navigate, between vertices in few steps even with very little knowledge of the graph itself. We discuss a different criterion for graphs being navigable in this sense, relating the neighbor selection of a vertex with the hitting probability of routed walks. In several models starting from both discrete and continuous settings, this can be showed to lead to graphs with the desired properties. It also leads directly to a evolutionary model for the creation of similar graphs by the stepwise rewiring of the edges, and we conjecture, supported by simulations, that these too are navigable.

http://arXiv.org/abs/math/0702325
http://front.math.ucdavis.edu/math.PR/0702325 (alternate)

5241. Continuity in law with respect to the Hurst parameter of the local time of the fractional Brownian motion

Author(s): Maria Jolis and No\`elia Viles

Abstract: We give a result of stability in law of the local time of the fractional Brownian motion with respect to small perturbations of the Hurst parameter. Concretely, we prove that the law (in the space of continuous functions) of the local time of the fractional Brownian motion with Hurst parameter $H$ converges weakly to that of the local time of $B^{H_0}$, when $H$ tends to $H_0$.

http://arXiv.org/abs/math/0702330
http://front.math.ucdavis.edu/math.PR/0702330 (alternate)

5242. Tightness conditions for polymer measures

Author(s): Francesco Caravenna and Giambattista Giacomin and Lorenzo Zambotti

Abstract: We give sufficient conditions for tightness in the space C([0,1]) for sequences of probability measures which enjoy a suitable decoupling between zero level set and excursions. Applications of our results are given in the context of (homogeneous, periodic and disordered) random walk models for polymers and interfaces.

http://arXiv.org/abs/math/0702331
http://front.math.ucdavis.edu/math.PR/0702331 (alternate)

5243. A note on equilibrium Glauber and Kawasaki dynamics for fermion point processes

Author(s): E. Lytvynov and N. Ohlerich

Abstract: We construct two types of equilibrium dynamics of infinite particle systems in a locally compact Polish space $X$, for which certain fermion point processes are invariant. The Glauber dynamics is a birth-and-death process in $X$, while in the case of the Kawasaki dynamics interacting particles randomly hop over $X$. We establish conditions on generators of both dynamics under which corresponding conservative Markov processes exist.

http://arXiv.org/abs/math/0702338
http://front.math.ucdavis.edu/math.PR/0702338 (alternate)

5244. Multiplicative free Convolution and Information-Plus-Noise Type Matrices

Author(s): {\O}yvind Ryan and M\'erouane Debbah

Abstract: Free probability and random matrix theory has shown to be a fruitful combination in many fields of research, such as digital communications, nuclear physics and mathematical finance. The link between free probability and eigenvalue distributions of random matrices will be strengthened further in this paper. It will be shown how the concept of multiplicative free convolution can be used to express known results for eigenvalue distributions of a type of random matrices called Information-Plus-Noise matrices. The result is proved in a free probability framework, and some new results, useful for problems related to free probability, are presented in this context. The connection between free probability and estimators for covariance matrices is also made through the notion of free deconvolution.

http://arXiv.org/abs/math/0702342
http://front.math.ucdavis.edu/math.PR/0702342 (alternate)

5245. Law of Large Numbers and Central Limit Theorem under Nonlinear Expectations

Author(s): Shige Peng

Abstract: The law of large numbers (LLN) and central limit theorem (CLT) are long and widely been known as two fundamental results in probability theory. Recently problems of model uncertainties in statistics, measures of risk and superhedging in finance motivated us to introduce, in [4] and [5] (see also [2], [3] and references herein), a new notion of sublinear expectation, called \textquotedblleft% $G$-expectation\textquotedblright, and the related \textquotedblleft$G$-normal distribution\textquotedblright from which we were able to define G-Brownian motion as well as the corresponding stochastic calculus. The notion of G-normal distribution plays the same important rule in the theory of sublinear expectation as that of normal distribution in the classic probability theory. It is then natural and interesting to ask if we have the corresponding LLN and CLT under a sublinear expectation and, in particular, if the corresponding limit distribution of the CLT is a G-normal distribution. This paper gives an affirmative answer. The proof of our CLT is short since we borrow a deep interior estimate of fully nonlinear PDE in [6] which extended a profound result of [1] (see also [7]) to parabolic PDEs. The assumptions of our LLN and CLT can be still improved. But the discovered phenomenon plays the same important rule in the theory of nonlinear expectation as that of the classical LLN and CLT in classic probability theory.

http://arXiv.org/abs/math/0702358
http://front.math.ucdavis.edu/math.PR/0702358 (alternate)

5246. Smoothness of density for solutions to stochastic differential equations with jumps

Author(s): T.R.Cass

Abstract: We consider a solution to a generic stochastic differential equation with jumps and show that for each time the marginal law of the solution has an infinitely differentiable density with respect to Lebesgue measure under a uniform version of Hoermanders conditions. Our results are proved subject to some restrictions on the rate of growth of the jump measure near zero and are accomplished using developments of traditional arguments in Malliavin calculus. A key ingredient in our proof is a generalisation of Norris semimartingale inequality to discontinuous semimartingales. Unlike previous work, our results extend beyond the case finite activity jump processes.

http://arXiv.org/abs/math/0702364
http://front.math.ucdavis.edu/math.PR/0702364 (alternate)

5247. A population model for $\Lambda$-coalescents with neutral mutations

Author(s): Andreas Nordvall Lager{\aa}s

Abstract: Bertoin and Le Gall (2003) introduced a certain probability measure valued Markov process that describes the evolution of a population, such that a sample from this population would exhibit a genealogy given by the so-called $\Lambda$-coalescent, or coalescent with multiple collisions, introduced independently by Pitman (1999) and Sagitov (1999). We show how this process can be extended to the case where lineages can experience mutations. Regenerative compositions enter naturally into this model, which is somewhat surprising, considering a negative result by M{\"o}hle (2007).

http://arXiv.org/abs/math/0702367
http://front.math.ucdavis.edu/math.PR/0702367 (alternate)

5248. Integral Equations in the Theory of Levy Processes

Author(s): Lev Sakhnovich

Abstract: In this article we consider the Levy processes and the corresponding semigroup. We represent the generator of this semigroup in a convolution form. Using the obtained convolution form and the theory of integral equations we investigate the properties of a wide class of Levy processes (potential, quasi-potential, the probability of the Levy process remaining within the given domain, long time behavior, stable processes). We analyze in detail a number of concrete examples of the Levy processes (stable processes, the variance damped Levy processes, the variance gamma processes, the normal Gaussian process, the Meixner process, the compound Poisson process).

http://arXiv.org/abs/math/0702378
http://front.math.ucdavis.edu/math.PR/0702378 (alternate)

5249. On the Circular Law

Author(s): F. G\"otze and A. Tikhomirov

Abstract: We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent real entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries. We assume however that the entries have sub-Gaussian tails or are sparsely non-zero.

http://arXiv.org/abs/math/0702386
http://front.math.ucdavis.edu/math.PR/0702386 (alternate)

5250. Stationary flows and uniqueness of invariant measures

Author(s): Francois Baccelli and Takis Konstantopoulos

Abstract: In this short paper, we consider a quadruple $(\Omega, \AA, \theta, \mu)$,where $\AA$ is a $\sigma$-algebra of subsets of $\Omega$, and $\theta$ is a measurable bijection from $\Omega$ into itself that preserves the measure $\mu$. For each $B \in \AA$, we consider the measure $\mu_B$ obtained by taking cycles (excursions) of iterates of $\theta$ from $B$. We then derive a relation for $\mu_B$ that involves the forward and backward hitting times of $B$ by the trajectory $(\theta^n \omega, n \in \Z)$ at a point $\omega \in \Omega$. Although classical in appearance, its use in obtaining uniqueness of invariant measures of various stochastic models seems to be new. We apply the concept to countable Markov chains and Harris processes.

http://arXiv.org/abs/math/0702391
http://front.math.ucdavis.edu/math.PR/0702391 (alternate)

5251. Majority bootstrap percolation on the hypercube

Author(s): J\'ozsef Balogh and B\'ela Bollob\'as and Robert Morris

Abstract: In majority bootstrap percolation on a graph G, an infection spreads according to the following deterministic rule: if at least half of the neighbours of a vertex v are already infected, then v is also infected, and infected vertices remain infected forever. Percolation occurs if eventually every vertex is infected. The elements of the set of initially infected vertices, A \subset V(G), are normally chosen independently at random, each with probability p, say. This process has been extensively studied on the sequence of torus graphs [n]^d, for n = 1,2,..., where d = d(n) is either fixed or a very slowly growing function of n. For example, Cerf and Manzo showed that the critical probability is o(1) if d(n) < log*(n), i.e., if p = p(n) is bounded away from zero then the probability of percolation on [n]^d tends to one as n goes to infinity. In this paper we study the case when the growth of d to infinity is not excessively slow; in particular, we show that the critical probability is 1/2 + o(1) if d > (loglog(n))^2 logloglog(n), and give much stronger bounds in the case that G is the hypercube, [2]^d.

http://arXiv.org/abs/math/0702373
http://front.math.ucdavis.edu/math.CO/0702373 (alternate)

5252. Bergman kernels and weighted equilibrium measures of C^n

Author(s): Robert Berman

Abstract: We obtain various convergence results for the Bergman kernel of the Hilbert space of all polynomials in \C^{n} of total degree at most k, equipped with a weighted norm. The weight function is assumed to be a smooth function in \C^{n} which grows faster than the logarithm of the squared distance function. The convergence is studied in the large k limit and is expressed in terms of the global equilibrium potential associated to the weight function, as well as in terms of the Monge-Ampere measure of the weight function itself on a certain bounded support set S. These results apply directly to the study of the distribution of zeroes of random polynomials and of the eigenvalues of random normal matrices.

http://arXiv.org/abs/math/0702357
http://front.math.ucdavis.edu/math.CV/0702357 (alternate)

5253. On the number of minima of a random polynomial

Author(s): Jean-Pierre Dedieu and Gregorio Malajovich

Abstract: We give an upper bound in O(d ^((n+1)/2)) for the number of critical points of a normal random polynomial. The number of minima (resp. maxima) is in O(d^((n+1)/2)) P_n, where P_n is the (unknown) measure of the set of symmetric positive matrices in the Gaussian Orthogonal Ensemble GOE(n). Finally, we give a closed form expression for the number of maxima (resp. minima) of a random univariate polynomial, in terms of hypergeometric functions.

http://arXiv.org/abs/math/0702360
http://front.math.ucdavis.edu/math.NA/0702360 (alternate)

5254. Diffusion approximations for controlled stochastic networks: An asymptotic bound for the value function

Author(s): Amarjit Budhiraja and Arka Prasanna Ghosh

Abstract: We consider the scheduling control problem for a family of unitary networks under heavy traffic, with general interarrival and service times, probabilistic routing and infinite horizon discounted linear holding cost. A natural nonanticipativity condition for admissibility of control policies is introduced. The condition is seen to hold for a broad class of problems. Using this formulation of admissible controls and a time-transformation technique, we establish that the infimum of the cost for the network control problem over all admissible sequencing control policies is asymptotically bounded below by the value function of an associated diffusion control problem (the Brownian control problem). This result provides a useful bound on the best achievable performance for any admissible control policy for a wide class of networks.

http://arXiv.org/abs/math/0702402
http://front.math.ucdavis.edu/math.PR/0702402 (alternate)

5255. Functional inequalities and uniqueness of the Gibbs measure -- from log-Sobolev to Poincar\'{e}

Author(s): Pierre-Andr\'{e} Zitt (MODAL'X)

Abstract: In a statistical mechanics model with unbounded spins, we prove uniqueness of the Gibbs measure under various assumptions on finite volume functional inequalities. We follow the approach of G. Royer (1999) and obtain uniqueness by showing convergence properties of a Glauber-Langevin dynamics. The result was known when the measures on the box $[-n,n]^d$ (with free boundary conditions) satisfied the same logarithmic Sobolev inequality. We generalize this in two directions: either the constants may be allowed to grow sub-linearly in the diameter, or we may suppose a weaker inequality than log-Sobolev, but stronger than Poincar\'{e}. We conclude by giving a heuristic argument showing that this could be the right inequalities to look at.

http://arXiv.org/abs/math/0702403
http://front.math.ucdavis.edu/math.PR/0702403 (alternate)

5256. Bounded solutions to backward SDE's with jumps for utility optimization and indifference hedging

Author(s): Dirk Becherer

Abstract: We prove results on bounded solutions to backward stochastic equations driven by random measures. Those bounded BSDE solutions are then applied to solve different stochastic optimization problems with exponential utility in models where the underlying filtration is noncontinuous. This includes results on portfolio optimization under an additional liability and on dynamic utility indifference valuation and partial hedging in incomplete financial markets which are exposed to risk from unpredictable events. In particular, we characterize the limiting behavior of the utility indifference hedging strategy and of the indifference value process for vanishing risk aversion.

http://arXiv.org/abs/math/0702405
http://front.math.ucdavis.edu/math.PR/0702405 (alternate)

5257. The Choquet-Deny theorem and distal properties of totally disconnected locally compact groups of polynomial growth

Author(s): W. Jaworski and C. R. E. Raja

Abstract: We obtain sufficient and necessary conditions for the Choquet-Deny theorem to hold in the class of compactly generated totally disconnected locally compact groups of polynomial growth, and in a larger class of totally disconnected generalized $\ov{FC}$-groups. The following conditions turn out to be equivalent when $G$ is a metrizable compactly generated totally disconnected locally compact group of polynomial growth: (i) the Choquet-Deny theorem holds for $G$; (ii) the group of inner automorphisms of $G$ acts distally on $G$; (iii) every inner automorphism of $G$ is distal; (iv) the contraction subgroup of every inner automorphism of $G$ is trivial; (v) $G$ is a SIN group. We also show that for every probability measure $\mu$ on a totally disconnected compactly generated locally compact second countable group of polynomial growth, the Poisson boundary is a homogeneous space of $G$, and that it is a compact homogeneous space when the support of $\mu$ generates $G$.

http://arXiv.org/abs/math/0702407
http://front.math.ucdavis.edu/math.PR/0702407 (alternate)

5258. Market free lunch and large financial markets

Author(s): Irene Klein

Abstract: The main result of the paper is a version of the fundamental theorem of asset pricing (FTAP) for large financial markets based on an asymptotic concept of no market free lunch for monotone concave preferences. The proof uses methods from the theory of Orlicz spaces. Moreover, various notions of no asymptotic arbitrage are characterized in terms of no asymptotic market free lunch; the difference lies in the set of utilities. In particular, it is shown directly that no asymptotic market free lunch with respect to monotone concave utilities is equivalent to no asymptotic free lunch. In principle, the paper can be seen as the large financial market analogue of [Math. Finance 14 (2004) 351--357] and [Math. Finance 16 (2006) 583--588].

http://arXiv.org/abs/math/0702409
http://front.math.ucdavis.edu/math.PR/0702409 (alternate)

5259. Separation cut-offs for birth and death chains

Author(s): Persi Diaconis and Laurent Saloff-Coste

Abstract: This paper gives a necessary and sufficient condition for a sequence of birth and death chains to converge abruptly to stationarity, that is, to present a cut-off. The condition involves the notions of spectral gap and mixing time. Y. Peres has observed that for many families of Markov chains, there is a cut-off if and only if the product of spectral gap and mixing time tends to infinity. We establish this for arbitrary birth and death chains in continuous time when the convergence is measured in separation and the chains all start at 0.

http://arXiv.org/abs/math/0702411
http://front.math.ucdavis.edu/math.PR/0702411 (alternate)

5260. Harris recurrence of Metropolis-within-Gibbs and trans-dimensional Markov chains

Author(s): Gareth O. Roberts and Jeffrey S. Rosenthal

Abstract: A $\phi$-irreducible and aperiodic Markov chain with stationary probability distribution will converge to its stationary distribution from almost all starting points. The property of Harris recurrence allows us to replace ``almost all'' by ``all,'' which is potentially important when running Markov chain Monte Carlo algorithms. Full-dimensional Metropolis--Hastings algorithms are known to be Harris recurrent. In this paper, we consider conditions under which Metropolis-within-Gibbs and trans-dimensional Markov chains are or are not Harris recurrent. We present a simple but natural two-dimensional counter-example showing how Harris recurrence can fail, and also a variety of positive results which guarantee Harris recurrence. We also present some open problems. We close with a discussion of the practical implications for MCMC algorithms.

http://arXiv.org/abs/math/0702412
http://front.math.ucdavis.edu/math.PR/0702412 (alternate)

5261. Sensitivity analysis of utility-based prices and risk-tolerance wealth processes

Author(s): Dmitry Kramkov and Mihai S\^{{\i}}rbu

Abstract: In the general framework of a semimartingale financial model and a utility function $U$ defined on the positive real line, we compute the first-order expansion of marginal utility-based prices with respect to a ``small'' number of random endowments. We show that this linear approximation has some important qualitative properties if and only if there is a risk-tolerance wealth process. In particular, they hold true in the following polar cases: \begin{tabular}@p97mm@ for any utility function $U$, if and only if the set of state price densities has a greatest element from the point of view of second-order stochastic dominance;for any financial model, if and only if $U$ is a power utility function ($U$ is an exponential utility function if it is defined on the whole real line). \end{tabular}

http://arXiv.org/abs/math/0702413
http://front.math.ucdavis.edu/math.PR/0702413 (alternate)

5262. Central limit theorem for the on-line nearest-neighbour graph

Author(s): Andrew R. Wade

Abstract: The on-line nearest-neighbour graph on a sequence of uniform random points in $(0,1)^d$ ($d \in \N$) joins each point after the first to its nearest neighbour amongst its predecessors. For the total power-weighted edge length of this graph, with weight exponent $\alpha \in (0,d/2)$, we prove a central limit theorem (in the large-sample limit), including an expression for the limiting variance. In contrast, we give a convergence result (with no scaling) for $\alpha > d/2$. Both these results extend previous work. We also make some progress in the critical case $\alpha=d/2$.

http://arXiv.org/abs/math/0702414
http://front.math.ucdavis.edu/math.PR/0702414 (alternate)

5263. The mean, variance and limiting distribution of two statistics sensitive to phylogenetic tree balance

Author(s): Michael G. B. Blum and Olivier Fran\c{c}ois and Svante Janson

Abstract: For two decades, the Colless index has been the most frequently used statistic for assessing the balance of phylogenetic trees. In this article, this statistic is studied under the Yule and uniform model of phylogenetic trees. The main tool of analysis is a coupling argument with another well-known index called the Sackin statistic. Asymptotics for the mean, variance and covariance of these two statistics are obtained, as well as their limiting joint distribution for large phylogenies. Under the Yule model, the limiting distribution arises as a solution of a functional fixed point equation. Under the uniform model, the limiting distribution is the Airy distribution. The cornerstone of this study is the fact that the probabilistic models for phylogenetic trees are strongly related to the random permutation and the Catalan models for binary search trees.

http://arXiv.org/abs/math/0702415
http://front.math.ucdavis.edu/math.PR/0702415 (alternate)

5264. Existence of optimal controls for singular control problems with state constraints

Author(s): Amarjit Budhiraja and Kevin Ross

Abstract: We establish the existence of an optimal control for a general class of singular control problems with state constraints. The proof uses weak convergence arguments and a time rescaling technique. The existence of optimal controls for Brownian control problems \citehar, associated with a broad family of stochastic networks, follows as a consequence.

http://arXiv.org/abs/math/0702418
http://front.math.ucdavis.edu/math.PR/0702418 (alternate)

5265. Stationarity and geometric ergodicity of a class of nonlinear ARCH models

Author(s): Youssef Sa\"{{\i}}di and Jean-Michel Zako\"{{\i}}an

Abstract: A class of nonlinear ARCH processes is introduced and studied. The existence of a strictly stationary and $\beta$-mixing solution is established under a mild assumption on the density of the underlying independent process. We give sufficient conditions for the existence of moments. The analysis relies on Markov chain theory. The model generalizes some important features of standard ARCH models and is amenable to further analysis.

http://arXiv.org/abs/math/0702419
http://front.math.ucdavis.edu/math.PR/0702419 (alternate)

5266. Corrections and acknowledgment for ``Local limit theory and large deviations for supercritical branching processes''

Author(s): P. E. Ney and Anand N. Vidyashankar

Abstract: Corrections and acknowledgment for ``Local limit theory and large deviations for supercritical branching processes'' [math.PR/0407059]

http://arXiv.org/abs/math/0702421
http://front.math.ucdavis.edu/math.PR/0702421 (alternate)

5267. Correction. Error estimates for binomial approximations of game options

Author(s): Yuri Kifer

Abstract: Correction for Error estimates for binomial approximations of game options [math.PR/0607123]

http://arXiv.org/abs/math/0702423
http://front.math.ucdavis.edu/math.PR/0702423 (alternate)

5268. Network-based analysis of stochastic SIR epidemic models with random and proportionate mixing

Author(s): Eben Kenah and James Robins

Abstract: In this paper, we outline the theory of percolation networks and their use in the analysis of stochastic epidemic models on undirected contact networks. We then show how the same theory can be used to analyze epidemic models with random mixing. In the percolation network for a random-mixing model, undirected edges disappear in the limit of a large population, so the percolation network is purely directed. In a series of simulations, we show that percolation networks accurately predict the mean outbreak size and probability and final size of an epidemic for a variety of epidemic models in homogeneous and heterogeneous populations. Finally, we show conditions under which percolation network models are equivalent to branching processes and use percolation networks to re-derive several classical results from different areas of infectious disease epidemiology. In an appendix, we show how percolation networks can be defined for any time-homogeneous stochastic epidemic model. We conclude that the theory of percolation on semi-directed networks provides a very general framework for the analysis of stochastic SIR epidemic models in closed populations, which are an important part of theoretical infectious disease epidemiology.

http://arXiv.org/abs/q-bio/0702027
http://front.math.ucdavis.edu/q-bio.QM/0702027 (alternate)

5269. Limiting shapes for deterministic internal growth models

Author(s): Anne Fey and Frank Redig

Abstract: We study the rotor router model and two deterministic sandpile models. For the rotor router model in $\mathbb{Z}^d$, Levine and Peres proved that the limiting shape of the growth cluster is a sphere. For the other two models, only bounds in dimension 2 are known. A unified approach for these models with a new parameter $h$ (the initial number of particles at each site), allows to prove a number of new limiting shape results in any dimension $d \geq 1$. For the rotor router model, the limiting shape is a sphere for all values of $h$. For one of the sandpile models, and $h=2d-2$ (the maximal value), the limiting shape is a cube. For both sandpile models, the limiting shape is a sphere in the limit $h \to -\infty$. Finally, we prove that the rotor router shape contains a diamond, which is a new result even in the case studied by Levine and Peres.

http://arXiv.org/abs/math/0702450
http://front.math.ucdavis.edu/math.PR/0702450 (alternate)

5270. Convexity theory for the term structure equation

Author(s): Erik Ekstrom and Johan Tysk

Abstract: We study convexity and monotonicity properties for prices of bonds and bond options when the short rate is modeled by a diffusion process. We provide conditions under which convexity of the price in the short rate is guaranteed. Under these conditions the price is decreasing in the drift and increasing in the volatility of the short rate. We also study convexity properties of the logarithm of the price.

http://arXiv.org/abs/math/0702435
http://front.math.ucdavis.edu/math.AP/0702435 (alternate)

5271. Some applications and methods of large deviations in finance and insurance

Author(s): Huyen Pham (PMA)

Abstract: In these notes, we present some methods and applications of large deviations to finance and insurance. We begin with the classical ruin problem related to the Cramer's theorem and give en extension to an insurance model with investment in stock market. We then describe how large deviation approximation and importance sampling are used in rare event simulation for option pricing. We finally focus on large deviations methods in risk management for the estimation of large portfolio losses in credit risk and portfolio performance in market investment.

http://arXiv.org/abs/math/0702473
http://front.math.ucdavis.edu/math.PR/0702473 (alternate)

5272. A note on percolation on \Z^d: isoperimetric profile via exponential cluster repulsion

Author(s): Gabor Pete

Abstract: We show that for all p>p_c(\Z^d) percolation parameters, the probability that the cluster of the origin is finite but has at least t vertices at distance one from the infinite cluster is exponentially small in t. Then we use this to give a very short proof of the important fact that the isoperimetric profile of the infinite cluster basically coincides with the profile of the original lattice. This implies for instance that simple random walk on the largest cluster of a finite box [-n,n]^d with high probability has L^\infty-mixing time \Theta(n^2), and that the heat kernel (return probability) on the infinite cluster a.s. decays like p_n(o,o)=O(n^{-d/2}). Versions of these results have been proven by Benjamini and Mossel (2003), Mathieu and Remy (2004), Barlow (2004) and Rau (2006). We also give a short proof of a theorem of Angel, Benjamini, Berger and Peres (2006): the infinite percolation cluster of a wedge in \Z^3 is a.s. transient whenever the wedge itself is transient.

http://arXiv.org/abs/math/0702474
http://front.math.ucdavis.edu/math.PR/0702474 (alternate)

5273. Central Limit Theorem for a Class of Relativistic Diffusions

Author(s): J\"{u}rgen Angst (IRMA) and Jacques Franchi (IRMA)

Abstract: Two similar Minkowskian diffusions have been considered, on one hand by Barbachoux, Debbasch, Malik and Rivet ([BDR1], [BDR2], [BDR3], [DMR], [DR]), and on the other hand by Dunkel and H\"{a}nggi ([DH1], [DH2]). We address here the question, asked in ([DH1], [DH2]), of the asymptotic behaviour of the variance of such diffusions. More generally, we establish a central limit theorem for a class of Minkowskian diffusions, to which the two above ones belong. As a consequence, we correct a partially wrong guess in [DH1].

http://arXiv.org/abs/math/0702481
http://front.math.ucdavis.edu/math.PR/0702481 (alternate)

5274. Scaling limits for gradient systems in random environment

Author(s): P. Goncalves and M.D. Jara

Abstract: For interacting particle systems that satisfies the gradient condition, the hydrodynamic limit and the equilibrium fluctuations are well known. We prove that under the presence of a symmetric random environment, these scaling limits also hold for almost every choice of the environment, with homogenized coefficients that does not depend on the particular realization of the random environment.

http://arXiv.org/abs/math/0702513
http://front.math.ucdavis.edu/math.PR/0702513 (alternate)

5275. First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation $\partial/\partial t = \pm\partial^N/\partial x^N$

Author(s): Aim\'e Lachal

Abstract: Consider the high-order heat-type equation $\partial u/\partial t=\pm\partial^N u/\partial x^N$ for an integer $N>2$ and introduce the related Markov pseudo-process $(X(t))_{t\ge 0}$. In this paper, we study several functionals related to $(X(t))_{t\ge 0}$: the maximum $M(t)$ and minimum $m(t)$ up to time $t$; the hitting times $\tau_a^+$ and $\tau_a^-$ of the half lines $(a,+\infty)$ and $(-\infty,a)$ respectively. We provide explicit expressions for the distributions of the vectors $(X(t),M(t))$ and $(X(t),m(t))$, as well as those of the vectors $(\tau_a^+,X(\tau_a^+))$ and $(\tau_a^-,X(\tau_a^-))$.

http://arXiv.org/abs/math/0702541
http://front.math.ucdavis.edu/math.PR/0702541 (alternate)

5276. Dynamics for the Brownian web and the erosion flow

Author(s): Chris Howitt and Jon Warren

Abstract: The Brownian web is a random object that occurs as the scaling limit of an infinite system of coalescing random walks. Perturbing this system of random walks by, independently at each point in space-time, resampling the random walk increments, leads to some natural dynamics. In this paper we consider the corresponding dynamics for the Brownian web. In particular, pairs of coupled Brownian webs are studied, where the second web is obtained from the first by perturbing according to these dynamics. A stochastic flow of kernels, which we call the erosion flow, is obtained via a filtering construction from such coupled Brownian webs, and the N-point motions of this flow of kernels are identified.

http://arXiv.org/abs/math/0702542
http://front.math.ucdavis.edu/math.PR/0702542 (alternate)

5277. Extreme Points of the Convex Set of Joint Probability Distributions with Fixed Marginals

Author(s): K. R. Parthasarathy

Abstract: By using a quantum probabilistic approach we obtain a description of the extreme points of the convex set of all joint probability distributions on the product of two standard Borel spaces with fixed marginal distributions.

http://arXiv.org/abs/math/0702544
http://front.math.ucdavis.edu/math.PR/0702544 (alternate)

5278. QQ plots, Random sets and data from a heavy tailed distribution

Author(s): Bikramjit Das and Sidney I. Resnick

Abstract: The QQ plot is a commonly used technique for informally deciding whether a univariate random sample of size n comes from a specified distribution F. The QQ plot graphs the sample quantiles against the theoretical quantiles of F and then a visual check is made to see whether or not the points are close to a straight line. For a location and scale family of distributions, the intercept and slope of the straight line provide estimates for the shift and scale parameters of the distribution respectively. Here we consider the set S_n of points forming the QQ plot as a random closed set in R^2. We show that under certain regularity conditions on the distribution F, S_n converges in probability to a closed, non-random set. In the heavy tailed case where 1-F is a regularly varying function, a similar result can be shown but a modification is necessary to provide a statistically sensible result since typically F is not completely known.

http://arXiv.org/abs/math/0702551
http://front.math.ucdavis.edu/math.PR/0702551 (alternate)

5279. Variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points

Author(s): Tomasz Schreiber and Joseph E. Yukich

Abstract: We show that the random point measures induced by vertices in the convex hull of a Poisson sample on the unit ball, when properly scaled and centered, converge to those of a mean zero Gaussian field. We establish limiting variance and covariance asymptotics in terms of the density of the Poisson sample. Similar results hold for the point measures induced by the maximal points in a Poisson sample. The approach involves introducing a generalized spatial birth growth process allowing for cell overlap.

http://arXiv.org/abs/math/0702553
http://front.math.ucdavis.edu/math.PR/0702553 (alternate)

5280. An improved method for model selection based on Information Criteria

Author(s): Guilhem Coq (1) and Olivier Alata (2) and Marc Arnaudon (1) and Christian Olivier (2) ((1) Laboratoire de Math\'ematiques et Applications Poitiers France, (2) Laboratoire Signal Image et Communications Poitiers France)

Abstract: Information criteria are an appropriate and widely used tool for solving model selection problems. However, different ways to use them exist, each leading to a more or less precise approximation of the sought model. In this paper, we mainly present two methods of utilisation of information criteria : the classical one which is generally used and an alternative one, more precise but requiring a little more calculations. Those methods are compared on 1-D and 2-D autoregressive models ; we use a synthetized process for the 1-D case and texture images for the 2-D case. We also work with the original phi_beta criterion which includes all others usual criteria such as AIC, BIC, and phi.

http://arXiv.org/abs/math/0702540
http://front.math.ucdavis.edu/math.ST/0702540 (alternate)

5281. A stochastic Lagrangian proof of global existence of the Navier-Stokes equations for flows with small Reynolds number

Author(s): Gautam Iyer

Abstract: We consider the incompressible Navier-Stokes equations with spatially periodic boundary conditions. If the Reynolds number is small enough we provide an elementary short proof of the existence of global in time H\"older continuous solutions. Our proof is based on the stochastic Lagrangian formulation of the Navier-Stokes equations, and works in both the two and three dimensional situation.

http://arXiv.org/abs/math/0702506
http://front.math.ucdavis.edu/math.AP/0702506 (alternate)

5282. Large deviation estimates of the crossing probability for pinned Gaussian processes

Author(s): L. Caramellino and B. Pacchiarotti

Abstract: The paper deals with the asymptotic behavior of the bridge of a Gaussian process conditioned to stay in $n$ fixed points at $n$ fixed past instants. In particular, functional large deviation results are stated for small time. Several examples are considered: integrated or not fractional Brownian motion, $m$-fold integrated Brownian motion. As an application, the asymptotic behavior of the exit probability is studied and used for the practical purpose of the numerical computation, via Monte Carlo methods, of the hitting probability up to a given time.

http://arXiv.org/abs/math/0702573
http://front.math.ucdavis.edu/math.PR/0702573 (alternate)

5283. Well-posedness and invariant measures for HJM models with deterministic volatility and Levy noise

Author(s): Carlo Marinelli

Abstract: We give sufficient conditions for existence, uniqueness and ergodicity of invariant measures for Musiela's stochastic partial differential equation with deterministic volatility and a Hilbert space valued driving Levy noise. Conditions for the absence of arbitrage and for the existence of mild solutions are also discussed.

http://arXiv.org/abs/math/0702622
http://front.math.ucdavis.edu/math.PR/0702622 (alternate)

5284. The Riff-Shuffle Distribution is Unimodal

Author(s): S. Gerhold

Abstract: We show that the probability mass function of the riff-shuffle distribution, also known as the minimum negative binomial distribution, is unimodal, but in general not log-concave.

http://arXiv.org/abs/math/0702639
http://front.math.ucdavis.edu/math.PR/0702639 (alternate)

5285. Bilateral Canonical Cascades: Multiplicative Refinement Paths to Wiener's and Variant Fractional Brownian Limits

Author(s): Julien Barral and Benoit Mandelbrot

Abstract: The original density is 1 for $t\in (0,1)$, $b$ is an integer base ($b\geq 2$%), and $p\in (0,1)$ is a parameter. The first construction stage divides the unit interval into $b$ subintervals and multiplies the density in each subinterval by either 1 or -1 with the respective frequencies of $\frac{1% }{2}+\frac{p}{2}$ and ${1/2}-\frac{p}{2}$. It is shown that the resulting density can be renormalized so that, as $n\to \infty $ ($n$ being the number of iterations) the signed measure converges in some sense to a non-degenerate limit. If $H=1+\log_{b}$ $p>{1}/{2}$, hence $p>b^{{-1}/{% 2}}$, renormalization creates a martingale, the convergence is strong, and the limit shares the H\"{o}lder and Hausdorff properties of the fractional Brownian motion of exponent $H$. If $H\leq {1}/{2}$, hence $p\leq b^{{-1}/{2}%}$, this martingale does not converge. However, a different normalization can be applied, for $H\leq {1/2}$ to the martingale itself and for $H>% {1/2}$ to the discrepancy between the limit and a finite approximation. In all cases the resulting process is found to converge weakly to the Wiener Brownian motion, independently of $H$ and of $b$. Thus, to the usual additive paths toward Wiener measure, this procedure adds an infinity of multiplicative paths.

http://arXiv.org/abs/math/0702644
http://front.math.ucdavis.edu/math.PR/0702644 (alternate)

5286. A survey of conformally invariant measures on H^m(\delta)

Author(s): Doug Pickrell

Abstract: The universal covering of the group PSU(1,1) acts naturally on H^m(\delta), the space of holomorphic differentials of order m on the Poincare disk. The purpose of this paper is to survey, as broadly as I am able, the basic sources and examples of invariant measures for this action.

http://arXiv.org/abs/math/0702672
http://front.math.ucdavis.edu/math.PR/0702672 (alternate)

5287. AR and MA representation of partial autocorrelation functions, with applications

Author(s): Akihiko Inoue

Abstract: We prove a representation of the partial autocorrelation function (PACF), or the Verblunsky coefficients, of a stationary process in terms of the AR and MA coefficients. We apply it to show the asymptotic behaviour of the PACF. We also propose a new definition of short and long memory in terms of the PACF.

http://arXiv.org/abs/math/0702648
http://front.math.ucdavis.edu/math.SP/0702648 (alternate)

5288. Classical dilations \`a la Quantum Probability of Markov evolutions in discrete time

Author(s): M. Gregoratti

Abstract: We study the Classical Probability analogue of the dilations of a quantum dynamical semigroup in Quantum Probability. Given a (not necessarily homogeneous) Markov chain in discrete time in a finite state space E, we introduce a second system, an environment, and a deterministic invertible time-homogeneous global evolution of the system E with this environment such that the original Markov evolution of E can be realized by a proper choice of the initial random state of the environment. We also compare this dilations with the dilations of a quantum dynamical semigroup in Quantum Probability: given a classical Markov semigroup, we show that it can be extended to a quantum dynamical semigroup for which we can find a quantum dilation to a group of *-automorphisms admitting an invariant abelian subalgebra where this quantum dilation gives just our classical dilation.

http://arXiv.org/abs/math/0702690
http://front.math.ucdavis.edu/math.PR/0702690 (alternate)

5289. Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particle systems

Author(s): Tomasz Bojdecki and Luis G. Gorostiza and Anna Talarczyk

Abstract: In this paper we study three self-similar, long-range dependence, Gaussian processes. The first one, with covariance \int_0^{s\wedge t} u^a [(t-u)^b+(s-u)^b]du, parameters a>-1, -1

http://arXiv.org/abs/math/0702708
http://front.math.ucdavis.edu/math.PR/0702708 (alternate)

5290. Hitting Probabilities for Systems of Non-Linear Stochastic Heat Equations with Additive Noise

Author(s): Robert C. Dalang and Davar Khoshnevisan and and Eulalia Nualart

Abstract: We consider a system of $d$ coupled non-linear stochastic heat equations in spatial dimension 1 driven by $d$-dimensional additive space-time white noise. We establish upper and lower bounds on hitting probabilities of the solution $\{u(t, x)\}_{t \in \mathbb{R}_+, x \in [0, 1]}$, in terms of respectively Hausdorff measure and Newtonian capacity. We also obtain the Hausdorff dimensions of level sets and their projections. A result of independent interest is an anisotropic form of the Kolmogorov continuity theorem.

http://arXiv.org/abs/math/0702710
http://front.math.ucdavis.edu/math.PR/0702710 (alternate)

5291. Uniform in bandwidth consistency of conditional U-statistics

Author(s): J. Dony and D. M. Mason

Abstract: In 1991 Stute introduced a class of estimators called conditional U-statistics. They can be seen as a generalization of the Nadaraya-Watson estimator, and their strong pointwise consistency to the general regression function has been obtained in the same paper by Stute. Very recently, Gine and Mason introduced the notion of a local U-process, which generalizes that of a local empirical process, and obtained central limit theorems and laws of the iterated logarithm for this class. We apply the methods developed by Einmahl and Mason (2005) and Gine and Mason (2007a,b) to establish uniform in bandwidth consistency to the general regression function of the estimator proposed by Stute.

http://arXiv.org/abs/math/0702696
http://front.math.ucdavis.edu/math.ST/0702696 (alternate)

5292. On the volume of nodal sets for eigenfunctions of the Laplacian on the torus

Author(s): Zeev Rudnick and Igor Wigman

Abstract: We study the volume of nodal sets for eigenfunctions of the Laplacian on the standard torus in two or more dimensions. We consider a sequence of eigenvalues $4\pi^2\eigenvalue$ with growing multiplicity $\Ndim\to\infty$, and compute the expectation and variance of the volume of the nodal set with respect to a Gaussian probability measure on the eigenspaces. We show that the expected volume of the nodal set is $const \sqrt{\eigenvalue}$. Our main result is that the variance of the volume normalized by $\sqrt{\eigenvalue}$ is bounded by $O(1/\sqrt{\Ndim})$, so that the normalized volume has vanishing fluctuations as we increase the dimension of the eigenspace.

http://arXiv.org/abs/math-ph/0702081
http://front.math.ucdavis.edu/math-ph/0702081 (alternate)

5293. A Portfolio Decomposition Formula

Author(s): Traian A Pirvu and Ulrich G Haussmann

Abstract: This paper derives a portfolio decomposition formula when the agent maximizes utility of her wealth at some finite planning horizon. The financial market is complete and consists of multiple risky assets (stocks) plus a risk free asset. The stocks are modelled as exponential Brownian motions with drift and volatility being Ito processes. The optimal portfolio has two components: a myopic component and a hedging one. We show that the myopic component is robust with respect to stopping times. We employ the Clark-Haussmann formula to derive portfolio s hedging component.

http://arXiv.org/abs/math/0702726
http://front.math.ucdavis.edu/math.PR/0702726 (alternate)

5294. On Robust Utility Maximization

Author(s): Traian A Pirvu and Ulrich G Haussmann

Abstract: This paper studies the problem of optimal investment in incomplete markets, robust with respect to stopping times. We work on a Brownian motion framework and the stopping times are adapted to the Brownian filtration. Robustness can only be achieved for logartihmic utility, otherwise a cashflow should be added to the investor s wealth. The cashflow can be decomposed into the sum of an increasing and a decreasing process. The last one can be viewed as consumption. The first one is an insurance premium the agent has to pay.

http://arXiv.org/abs/math/0702727
http://front.math.ucdavis.edu/math.PR/0702727 (alternate)

5295. Matrix norms and rapid mixing for spin systems

Author(s): Martin Dyer and Leslie Ann Goldberg and Mark Jerrum

Abstract: We give a systematic development of the application of matrix norms to rapid mixing in spin systems. We show that rapid mixing of both random update Glauber dynamics and systematic scan Glauber dynamics occurs if any matrix norm of the associated dependency matrix is less than 1. We give improved analysis for the case in which the diagonal of the dependency matrix is 0 (as in heat bath dynamics). We apply the matrix norm methods to random update and systematic scan Glauber dynamics for colouring various classes of graphs. We give a general method for estimating a norm of a symmetric non-regular matrix. This leads to improved mixing times for any class of graphs which is hereditary and sufficiently sparse including several classes of degree-bounded graphs such as non-regular graphs, trees, planar graphs and graphs with given tree-width and genus.

http://arXiv.org/abs/math/0702744
http://front.math.ucdavis.edu/math.PR/0702744 (alternate)

5296. Asymptotics of the minimum manipulating coalition size for positional voting rules under IC behaviour

Author(s): Geoffrey Pritchard and Mark C. Wilson

Abstract: We consider the problem of manipulation of elections using positional voting rules under Impartial Culture voter behaviour. We consider both the logical possibility of coalitional manipulation, and the number of voters that must be recruited to form a manipulating coalition. It is shown that the manipulation problem may be well approximated by a very simple linear program in two variables. This permits a comparative analysis of the asymptotic (large-population) manipulability of the various rules. It is seen that the manipulation resistance of positional rules with 5 or 6 (or more) candidates is quite different from the more commonly analyzed 3- and 4-candidate cases.

http://arXiv.org/abs/math/0702752
http://front.math.ucdavis.edu/math.PR/0702752 (alternate)

5297. Classical dilations \`a la Hudson-Parthasarathy of Markov semigroups

Author(s): M. Gregoratti

Abstract: We study the Classical Probability analogue of the dilations of a quantum dynamical semigroup defined in Quantum Probability via quantum stochastic differential equations. Given a homogeneous Markov chain in continuous time in a finite state space E, we introduce a second system, an environment, and a deterministic invertible time-homogeneous global evolution of the system E with this environment such that the original Markov evolution of E can be realized by a proper choice of the initial random state of the environment. We also compare this dilations with the dilations of a quantum dynamical semigroup in Quantum Probability: given a classical Markov semigroup, we extend it to a proper quantum dynamical semigroup for which we can find a Hudson-Parthasarathy dilation which is itself an extension of our classical dilation.

http://arXiv.org/abs/math/0702784
http://front.math.ucdavis.edu/math.PR/0702784 (alternate)

5298. Further results on some singular linear stochastic differential equations

Author(s): Larbi Alili and Ching-Tang Wu

Abstract: A class of Volterra transforms, preserving the Wiener measure, with kernels of Goursat type is considered. We provide some results on the inverses of the associated Gramian matrices. These are applied to the study of a class of linear singular stochastic differential equations together with the corresponding decompositions of filtrations. The studied equations are viewed as non-canonical decompositions of some generalized bridges.

http://arXiv.org/abs/math/0702785
http://front.math.ucdavis.edu/math.PR/0702785 (alternate)

5299. Stochastic Hamiltonian dynamical systems

Author(s): Joan-Andreu L\'azaro-Cam\'{\i} and Juan-Pablo Ortega

Abstract: We use the global stochastic analysis tools introduced by P. A. Meyer and L. Schwartz to write down a stochastic generalization of the Hamilton equations on a Poisson manifold that, for exact symplectic manifolds, satisfy a natural critical action principle similar to the one encountered in classical mechanics. Several features and examples in relation with the solution semimartingales of these equations are presented.

http://arXiv.org/abs/math/0702787
http://front.math.ucdavis.edu/math.PR/0702787 (alternate)

5300. Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees

Author(s): Yueyun Hu (LAGA) and Zhan Shi (PMA)

Abstract: We establish a second-order almost sure limit theorem for the minimal position in a one-dimensional super-critical branching random walk, and also prove a martingale convergence theorem which answers a question of Biggins and Kyprianou (2005). Our method applies furthermore to the study of directed polymers on a disordered tree; in particular, we give a rigorous proof of a phase transition phenomenon for the partition function, described by Derrida and Spohn (1988).

http://arXiv.org/abs/math/0702799
http://front.math.ucdavis.edu/math.PR/0702799 (alternate)

5301. Orbital approach to microstate free entropy

Author(s): Fumio Hiai and Takuho Miyamoto and Yoshimichi Ueda

Abstract: Motivated by Voiculescu's liberation theory, we introduce the orbital free entropy $\chi_orb$ for non-commutative self-adjoint random variables (also for "hyperfinite random multivariables"). Besides its basic properties the relation of $\chi_orb$ with the usual free entropy $\chi$ is shown. Moreover, the dimension counterpart of $\chi_orb$ is discussed.

http://arXiv.org/abs/math/0702745
http://front.math.ucdavis.edu/math.OA/0702745 (alternate)

5302. Poisson process approximation: From Palm theory to Stein's method

Author(s): Louis H. Y. Chen and Aihua Xia

Abstract: This exposition explains the basic ideas of Stein's method for Poisson random variable approximation and Poisson process approximation from the point of view of the immigration-death process and Palm theory. The latter approach also enables us to define local dependence of point processes [Chen and Xia (2004)] and use it to study Poisson process approximation for locally dependent point processes and for dependent superposition of point processes.

http://arXiv.org/abs/math/0702820
http://front.math.ucdavis.edu/math.PR/0702820 (alternate)

5303. Price systems for markets with transaction costs and control problems for some finance problems

Author(s): Tzuu-Shuh Chiang and Shang-Yuan Shiu and Shuenn-Jyi Sheu

Abstract: In a market with transaction costs, the price of a derivative can be expressed in terms of (preconsistent) price systems (after Kusuoka (1995)). In this paper, we consider a market with binomial model for stock price and discuss how to generate the price systems. From this, the price formula of a derivative can be reformulated as a stochastic control problem. Then the dynamic programming approach can be used to calculate the price. We also discuss optimization of expected utility using price systems.

http://arXiv.org/abs/math/0702828
http://front.math.ucdavis.edu/math.PR/0702828 (alternate)

5304. Asymptotic arbitrage and num\'eraire portfolios in large financial markets

Author(s): Dmitry B. Rokhlin

Abstract: This paper deals with the notion of a large financial market and the concepts of asymptotic arbitrage and strong asymptotic arbitrage (both of the first kind), introduced by Yu.M. Kabanov and D.O. Kramkov. We show that the arbitrage properties of a large market are completely determined by the asymptotic behavior of the sequence of the num\'eraire portfolios, related to the small markets. The obtained criteria can be expressed in terms of contiguity, entire separation and Hellinger integrals, provided these notions are extended to sub-probability measures. As examples we consider market models on finite probability spaces, semimartingale and diffusion models. Also a discrete-time infinite horizon market model with one log-normal stock is examined.

http://arXiv.org/abs/math/0702849
http://front.math.ucdavis.edu/math.PR/0702849 (alternate)
stefano . iacus at unimi . it