Probability Abstracts 114

This document contains abstracts 9660-9932
from Jan-1-2010 to Feb-28-2010.
They have been mailed on Mar 1st, 2010.

9660. Multiplicative strong unimodality for positive stable laws

Author(s): Thomas Simon (LPP)

Abstract: It is known that real Non-Gaussian stable distributions are unimodal, not additive strongly unimodal, and multiplicative strongly unimodal in the symmetric case. By a theorem of Cuculescu-Theodorescu, the only remaining relevant situation for the multiplicative strong unimodality of stable laws is the one-sided. In this paper, we show that positive $\alpha-$stable laws are multiplicative strongly unimodal iff $\alpha\le 1/2.$

http://arxiv.org/abs/1002.4977

9661. Functional macroscopic behavior of weighted random ball model

Author(s): Jean-Christophe Breton and Clement Dombry

Abstract: We consider a generalization of the weighted random ball model. The model is driven by a random Poisson measure with a product heavy tailed intensity measure. Such a model typically represents the transmission of a network of stations with a fading effect. In a previous article, the authors proved the convergence of the finite-dimensional distributions of related generalized random fields under various scalings and in the particular case when the fading function is the indicator function of the unit ball. In this paper, tightness and functional convergence are investigated. Using suitable moment estimates, we prove functional convergences for some parametric classes of configurations under the so-called large ball scaling and intermediate ball scaling. Convergence in the space of distributions is also discussed.

http://arxiv.org/abs/1002.4985

9662. On the size of a finite vacant cluster of random interlacements with small intensity

Author(s): Augusto Teixeira

Abstract: In this paper we establish some properties of percolation for the vacant set of random interlacements, for d at least 5 and small intensity u. The model of random interlacements was first introduced by A.S. Sznitman in arXiv:0704.2560. It is known that, for small u, almost surely there is a unique infinite connected component in the vacant set left by the random interlacements at level u, see arXiv:0808.3344 and arXiv:0805.4106. We estimate here the distribution of the diameter and the volume of the vacant component at level u containing the origin, given that it is finite. This comes as a by-product of our main theorem, which proves a stretched exponential bound on the probability that the interlacement set separates two macroscopic connected sets in a large cube. As another application, we show that with high probability, the unique infinite connected component of the vacant set is `ubiquitous' in large neighborhoods of the origin.

http://arxiv.org/abs/1002.4995

9663. The largest component in an inhomogeneous random intersection graph with clustering

Author(s): Mindaugas Bloznelis

Abstract: Given b>0, integers n, m=bn and a probability measure Q on {0, 1,..., m}, consider the random intersection graph on the vertex set [n]={1, ..., n}, where i and j are declared adjacent whenever S(i) and S(j) intersect. Here S(1), ..., S(n) denote iid random subsets of [m] such that P(|S(i)|=k)=Q(k). For sparse random intersection graphs we establish a first order asymptotic for the order of the largest connected component N=n(1-Q(0))g+o(n) in probability. Here g is an average of nonextinction probabilities of a related multi-type Poisson branching process.

http://arxiv.org/abs/1002.4649

9664. The martingale approach to disorder irrelevance for pinning models

Author(s): Hubert Lacoin

Abstract: This paper presents a very simple and self-contained proof of disorder irrelevance for inhomogeneous pinning models with return exponent \alpha\in (0,1/2). We also give a new upper bound for the contact fraction of the disordered model at criticality.

http://arxiv.org/abs/1002.4753

9665. Stochastic monotonicity and duality for one-dimensional Markov processes

Author(s): Vassili Kolokoltsov

Abstract: Stochastic monotonicity and the related duality are well studied for one-dimensional diffusions and discrete Markov chains. In this note we extend the theory to arbitrary one-dimensional Markov Feller processes. This seems to be relevant in connection with the recent increase of interest to the analysis of general processes containing jumps, in particular in financial mathematics.

http://arxiv.org/abs/1002.4773

9666. Brownian semistationary processes and conditional full support

Author(s): Mikko S. Pakkanen

Abstract: In this note, we study the infinite-dimensional conditional laws of Brownian semistationary processes. Motivated by the fact that these processes are typically not semimartingales, we present sufficient conditions ensuring that a Brownian semistationary process has conditional full support, a property introduced by Guasoni, R\'asonyi, and Schachermayer [Ann. Appl. Probab., 18 (2008) pp. 491--520]. By the results of Guasoni, R\'asonyi, and Schachermayer, this property has two important implications. It ensures, firstly, that the process admits no free lunches under proportional transaction costs, and secondly, that it can be approximated pathwise (in the sup norm) by semimartingales that admit equivalent martingale measures.

http://arxiv.org/abs/1002.4774

9667. Transformations of one-dimensional Gibbs measures with infinite range interaction

Author(s): Frank Redig and Feijia Wang

Abstract: We study single-site stochastic and deterministic transforma- tions of one-dimensional Gibbs measures in the uniqueness regime with infinite-range interactions. We prove conservation of Gibbsianness and give quantitative estimates on the decay of the transformed potential. As examples, we consider exponentially decaying potentials, and potentials decaying as a power-law.

http://arxiv.org/abs/1002.4796

9668. On the neighborhood radius estimation in Variable-neighborhood Markov Random Fields

Author(s): Enza Orlandi and Eva Loecherbach

Abstract: We consider Markov Random Fields defined by finite-region conditional probabilities depending on a neighborhood of the region which changes with the boundary conditions. The formal definition of these models requires partitions of the set of configurations according to their projections on finite neighborhoods of each lattice site. Each of these projections is called a context for the site. This framework is a natural extension, to d-dimensional fields, of the notion of variable-length Markov chains introduced by Rissanen (1983) in his classical paper. We define an algorithm to estimate the radius of the smallest ball containing the context based on a realization of the field. We prove the consistency of this estimator when the Dobrushin uniqueness condition for the one point conditional probabilities holds. Our proofs are constructive and yield explicit upper bounds for the probability of wrong estimation of the radius of the context.

http://arxiv.org/abs/1002.4850

9669. Generically stable and smooth measures in NIP theories

Author(s): Ehud Hrushovski and Anand Pillay and Pierre Simon

Abstract: We study stable like behaviour in first order theories without the independence property. We introduce generically stable measures, give characterizatiions, and show their ubiquity. We also introduce generic compact domination. We also prove the approximate definability of arbitrary Borel probability measures on definable sets in the real and p-adic fields.

http://arxiv.org/abs/1002.4763

9670. Cumulants and convolutions via Abel polynomials

Author(s): E. Di Nardo and P. Petrullo and D. Senato

Abstract: We provide an unifying polynomial expression giving moments in terms of cumulants, and viceversa, holding in the classical, boolean and free setting. This is done by using a symbolic treatment of Abel polynomials. As a by-product, we show that in the free cumulant theory the volume polynomial of Pitman and Stanley plays the role of the complete Bell exponential polynomial in the classical theory. Moreover via generalized Abel polynomials we construct a new class of cumulants, including the classical, boolean and free ones, and the convolutions linearized by them. Finally, via an umbral Fourier transform, we state a explicit connection between boolean and free convolution.

http://arxiv.org/abs/1002.4803

9671. Asymptotic distribution of singular values of powers of random matrices

Author(s): Nikita Alexeev (Saint-Petersburg State University and Russia) and Friedrich G\"otze (University of Bielefeld, Germany), and Alexander Tikhomirov (Syktyvkar State University, Russia)

Abstract: Let $x$ be a complex random variable such that ${\E {x}=0}$, ${\E |x|^2=1}$, ${\E |x|^{4} < \infty}$. Let $x_{ij}$, $i,j \in \{1,2,...\}$ be independet copies of $x$. Let ${\Xb=(N^{-1/2}x_{ij})}$, $1\leq i,j \leq N$ be a random matrix. Writing $\Xb^*$ for the adjoint matrix of $\Xb$, consider the product $\Xb^m{\Xb^*}^m$ with some $m \in \{1,2,...\}$. The matrix $\Xb^m{\Xb^*}^m$ is Hermitian positive semi-definite. Let $\lambda_1,\lambda_2,...,\lambda_N$ be eigenvalues of $\Xb^m{\Xb^*}^m$ (or squared singular values of the matrix $\Xb^m$). In this paper we find the asymptotic distribution function \[ G^{(m)}(x)=\lim_{N\to\infty}\E{F_N^{(m)}(x)} \] of the empirical distribution function \[ {F_N^{(m)}(x)} = N^{-1} \sum_{k=1}^N {\mathbb{I}{\{\lambda_k \leq x\}}}, \] where $\mathbb{I} \{A\}$ stands for the indicator function of event $A$. The moments of $G^{(m)}$ satisfy \[ M^{(m)}_p=\int_{\mathbb{R}}{x^p dG^{(m)}(x)}=\frac{1}{mp+1}\binom{mp+p}{p}. \] In Free Probability Theory $M^{(m)}_p$ are known as Fuss--Catalan numbers. With $m=1$ our result turns to a well known result of Marchenko--Pastur 1967.

http://arxiv.org/abs/1002.4442

9672. Phi-entropy inequalities and Fokker-Planck equations

Author(s): Fran\c{c}ois Bolley (CEREMADE) and Ivan Gentil (CEREMADE)

Abstract: We present new $\Phi$-entropy inequalities for diffusion semigroups under the curvature-dimension criterion. They include the isoperimetric function of the Gaussian measure. Applications to the long time behaviour of solutions to Fokker-Planck equations are given.

http://arxiv.org/abs/1002.4478

9673. Nonlinear Expectations and Stochastic Calculus under Uncertainty

Author(s): Shige Peng

Abstract: In this book, we introduce a new approach of sublinear expectation to deal with the problem of probability and distribution model uncertainty. We a new type of (robust) normal distributions and the related central limit theorem under sublinear expectation. We also present a new type of Brownian motion under sublinear expectations and the related stochastic calculus of Ito's type. The results provide robust tools for the problem of probability model uncertainty arising from financial risk management, statistics and stochastic controls.

http://arxiv.org/abs/1002.4546

9674. Dynamics of the supermarket model

Author(s): I. M. MacPhee and M. V. Menshikov and M. Vachkovskaia

Abstract: We consider the long term behaviour of a Markov chain \xi(t) on \Z^N based on the N station supermarket model. Different routing policies for the supermarket model give different Markov chains. We show that for a general class of local routing policies, "join the least weighted queue" (JLW), the N one-dimensional components \xi_i(t) can be partitioned into disjoint clusters C_k. Within each cluster C_k the "speed" of each component \xi_j converges to a constant V_k and under certain conditions \xi is recurrent in shape on each cluster. To establish these results we have assembled methods from two distinct areas of mathematics, semi-martingale techniques used for showing stability of Markov chains together with the theory of optimal flows in networks. As corollaries to our main result we obtain the stability classification of the supermarket model under any JLW policy and can explicitly compute the C_k and V_k for any instance of the model and specific JLW policy.

http://arxiv.org/abs/1002.4570

9675. Heavy traffic on a controlled motorway

Author(s): F. P. Kelly and R. J. Williams

Abstract: Unlimited access to a motorway network can, in overloaded conditions, cause a loss of capacity. Ramp metering (signals on slip roads to control access to the motorway) can help avoid this loss of capacity. The design of ramp metering strategies has several features in common with the design of access control mechanisms in communication networks. Inspired by models and rate control mechanisms developed for Internet congestion control, we propose a Brownian network model as an approximate model for a controlled motorway and consider it operating under a proportionally fair ramp metering policy.We present an analysis of the performance of this model.

http://arxiv.org/abs/1002.4591

9676. Nonparametric Estimation and On-Line Prediction for General Stationary Ergodic Sources

Author(s): Joe Suzuki

Abstract: We proposed a learning algorithm for nonparametric estimation and on-line prediction for general stationary ergodic sources. We prepare histograms each of which estimates the probability as a finite distribution, and mixture them with weights to construct an estimator. The whole analysis is based on measure theory. The estimator works whether the source is discrete or continuous. If it is stationary ergodic, then the measure theoretically given Kullback-Leibler information divided by the sequence length $n$ converges to zero as $n$ goes to infinity. In particular, for continuous sources, the method does not require existence of a probability density function.

http://arxiv.org/abs/1002.4453

9677. On a new probabilistic representation for the solution of the heat equation

Author(s): Paolo Da Pelo and Alberto Lanconelli

Abstract: We obtain a new probabilistic representation for the solution of the heat equation in terms of a product for smooth random variables which is introduced and studied in this paper. This multiplication, expressed in terms of the Hida-Malliavin derivatives of the random variables involved, exhibits many useful properties which are to some extents opposite to some peculiar features of the Wick product.

http://arxiv.org/abs/1002.4269

9678. Crossing random walks and stretched polymers at weak disorder

Author(s): Dmitry Ioffe and Yvan Velenik

Abstract: We consider a model of a polymer in Z^{d+1}, constrained to join 0 and a hyperplane at distance N. The polymer is subject to a quenched non-negative random environment. Alternatively, the model describes crossing random walks in a random potential (see Chapter 5 of [Sznitman] for the original Brownian motion formulation). It was recently shown, by Flury and by Zygouras, that, in such a setting, the quenched and annealed free energies coincide in the limit N to infinity, when d is at least 3 and the temperature is sufficiently high. We first strengthen this result by proving that, under somewhat weaker assumptions on the distribution of disorder which, in particular, enable a small probability of traps, the ratio of quenched and annealed partition functions actually converges. We then conclude that, in this case, the polymer obeys a diffusive scaling, with the same diffusivity constant as the annealed model.

http://arxiv.org/abs/1002.4289

9679. Well-posedness and large deviation for degenerate SDEs with Sobolev coefficients

Author(s): Xicheng Zhang

Abstract: In this article we prove the existence and uniqueness for degenerate stochastic differential equations with Sobolev (possibly singular) drift and diffusion coefficients in a generalized sense. In particular, our result covers the classical DiPerna-Lions flows and, we also obtain the well-posedness for degenerate Fokker-Planck equations with irregular coefficients. Moreover, a large deviation principle of Freidlin-Wenzell type for this type of SDEs is established.

http://arxiv.org/abs/1002.4297

9680. Coupling time distribution asymptotics for some couplings of the Levy stochastic area

Author(s): Wilfrid S. Kendall

Abstract: We exhibit some explicit co-adapted couplings for n-dimensional Brownian motion and all its Levy stochastic areas. In the two-dimensional case we show how to derive exact asymptotics for the coupling time under various mixed coupling strategies, using Dufresne's formula for the distribution of exponential functionals of Brownian motion. This yields quantitative asymptotics for the distributions of random times required for certain simultaneous couplings of stochastic area and Brownian motion. The approach also applies to higher dimensions, but will then lead to upper and lower bounds rather than exact asymptotics.

http://arxiv.org/abs/1002.4348

9681. Weak disorder asymptotics in the stochastic mean-field model of distance

Author(s): Shankar Bhamidi and Remco van der Hofstad

Abstract: In the recent past, there has been a concerted effort to develop mathematical models for real-world networks and analyze various dynamics on these models. One particular problem of significant importance is to understand the effect of random edge lengths or costs on the geometry and flow transporting properties of the network. Two different regimes are of great interest, the weak disorder regime where optimality of a path is determined by the sum of edge weights on the path and the strong disorder regime where optimality of a path is determined by the maximal edge weight on the path. In the context of the stochastic mean-field model of distance, we provide the first mathematically tractable model of weak disorder and show that no transition occurs at finite temperature. Indeed we show that for all fixed finite temperatures, the number of edges on the minimal weight path (i.e the hopcount) is always $\Theta(\log{n})$ and satisfies a central limit theorem with asymptotic means and variances of order $\Theta(\log{n})$, with limiting constants expressible in terms of the Malthusian rate of growth and the mean of the stable-age distribution of the associated continuous-time branching process. More precisely, we take independent and identically distributed edge weights with distribution $E^s$ for some parameter $s>0$, where $E$ is an exponential random variable with mean 1. Then, the asymptotic mean and variance of the central limit theorem for the hopcount are $s\log{n}$ and $s^2 \log{n}$ respectively. We also find limiting distributional asymptotics for the value of the minimal weight path in terms of extreme value distributions, Cox processes and martingale limits of branching processes.

http://arxiv.org/abs/1002.4362

9682. Regularity partitions and the topology of graphons

Author(s): L\'aszl\'o Lov\'asz and Bal\'azs Szegedy

Abstract: We highlight a topological aspect of the graph limit theory. Graphons are limit objects for convergent sequences of dense graphs. We introduce the representation of a graphon on a unique metric space and we relate the dimension of this metric space to the size of regularity partitions. We prove that if a graphon has an excluded induced sub-bigraph then the underlying metric space is compact and has finite packing dimension. It implies in particular that such graphons have regularity partitions of polynomial size.

http://arxiv.org/abs/1002.4377

9683. A characterization of freeness by invariance under quantum spreading

Author(s): Stephen Curran

Abstract: We construct spaces of quantum increasing sequences, which give quantum families of maps in the sense of Soltan. We then introduce a notion of quantum spreadability for a sequence of noncommutative random variables, by requiring their joint distribution to be invariant under taking quantum subsequences. Our main result is a free analogue of a theorem of Ryll-Nardzewski: for an infinite sequence of noncommutative random variables, quantum spreadability is equivalent to free independence and identical distribution with respect to a conditional expectation.

http://arxiv.org/abs/1002.4390

9684. Psi-series method in random trees and moments of high orders

Author(s): Hua-Huai Chern and Hsien-Kuei Hwang and Conrado Mart\'inez

Abstract: An unusual and surprising expansion of the form \[ p_n = \rho^{-n-1}(6n +\tfrac{18}5+ \tfrac{336}{3125} n^{-5}+\tfrac{1008}{3125} n^{-6} +\text{smaller order terms}), \] as $n\to\infty$, is derived for the probability $p_n$ that two randomly chosen binary search trees are identical (in shape and in labels of all corresponding nodes). A quantity arising in the analysis of phylogenetic trees is also proved to have a similar asymptotic expansion. Our method of proof is new in the literature of discrete probability and analysis of algorithms, and based on the psi-series expansions for nonlinear differential equations. Such an approach is very general and applicable to many other problems involving nonlinear differential equations; many examples are discussed and several attractive phenomena are discovered.

http://arxiv.org/abs/1002.3859

9685. A sharper threshold for bootstrap percolation in two dimensions

Author(s): Janko Gravner and Alexander E. Holroyd and Robert Morris

Abstract: Two-dimensional bootstrap percolation is a cellular automaton in which sites become 'infected' by contact with two or more already infected nearest neighbors. We consider these dynamics, which can be interpreted as a monotone version of the Ising model, on an n x n square, with sites initially infected independently with probability p. The critical probability p_c is the smallest p for which the probability that the entire square is eventually infected exceeds 1/2. Holroyd determined the sharp first-order approximation: p_c \sim \pi^2/(18 log n) as n \to \infty. Here we sharpen this result, proving that the second term in the expansion is -(log n)^{-3/2+ o(1)}, and moreover determining it up to a poly(log log n)-factor.

http://arxiv.org/abs/1002.3881

9686. Almost sure asymptotics for the random binary search tree

Author(s): Matthew I. Roberts

Abstract: We consider a (random permutation model) binary search tree with n nodes and give asymptotics on the loglog scale for the height H_n and saturation level h_n of the tree as n\to\infty, both almost surely and in probability. We then consider the number F_n of particles at level H_n at time n, and show that F_n is unbounded almost surely.

http://arxiv.org/abs/1002.3896

9687. Parameter estimations for SPDEs with multiplicative fractional noise

Author(s): Igor Cialenco

Abstract: We study parameter estimation problem for diagonalizable parabolic stochastic partial differential equations driven by a multiplicative fractional noise with any Hurst parameter $H\in(0,1)$. Two classes of estimates are investigated: traditional maximum likelihood type estimates, and a new class called closed-form exact estimates. Finally several examples are discussed, including statistical inference for stochastic heat equation driven by a fractional Brownian motion.

http://arxiv.org/abs/1002.3911

9688. Large number of queues in tandem: Scaling properties under back-pressure algorithm

Author(s): Alexander Stolyar

Abstract: We consider a system with N unit-service-rate queues in tandem, with exogenous arrivals of rate lambda at queue 1, under a back-pressure (MaxWeight) algorithm: service at queue n is blocked unless its queue length is greater than that of next queue n+1. The question addressed is how steady-state queues scale as N goes to infinity. We show that the answer depends on whether lambda is below or above the critical value 1/4: in the former case queues remain uniformly stochastically bounded, while otherwise they grow to infinity. The problem is essentially reduced to the behavior of the system with infinite number of queues in tandem, which is studied using tools from interacting particle systems theory. In particular, the criticality of load 1/4 is closely related to the fact that this is the maximum possible flux (flow rate) of a stationary totally asymmetric simple exclusion process.

http://arxiv.org/abs/1002.3940

9689. Variations on the Berry-Esseen theorem

Author(s): Bo'az Klartag and Sasha Sodin

Abstract: We analyze the quality of the gaussian approximation to linear combinations of n independent, identically-distributed random variables with finite fourth moments. It turns out that there exist universal, simple linear combinations that perform better than the sum of the variables. We also investigate the case in which the random variables are independent, yet they are not necessarily identically distributed.

http://arxiv.org/abs/1002.3970

9690. Moments of Gamma type and the Brownian supremum process area

Author(s): Svante Janson

Abstract: We study positive random variables whose moments can be expressed by products and quotients of Gamma functions; this includes many standard distributions. General results are given on existence, series expansion and asymptotics of density functions. It is shown that the integral of the supremum process of Brownian motion has moments of this type, as well as a related random variable occuring in the study of hashing with linear displacement, and the general results are applied to these variables.

http://arxiv.org/abs/1002.4135

9691. A limit theorem for particle current in the symmetric exclusion process

Author(s): Alexander Vandenberg-Rodes

Abstract: Using the recently discovered strong negative dependence properties of the symmetric exclusion process, we derive general conditions for when the normalized current of particles between regions converges to the Gaussian distribution. The main novelty is that the results do not assume any translation invariance, and hold for most initial configurations.

http://arxiv.org/abs/1002.4148

9692. Particle systems with quasi-homogeneous initial states and their occupation time fluctuations

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

Abstract: Occupation time fluctuation limits of particle systems in R^d with independent motions (symmetric stable Levy process, with or without critical branching) have been studied assuming initial distributions given by Poisson random measures (homogeneous and some inhomogeneous cases). In this paper, with d=1 for simplicity, we extend previous results to a wide class of initial measures obeying a quasi-homogeneity property, which includes as special cases homogeneous Poisson measures and many deterministic measures (simple example: one atom at each point of Z), by means of a new unified approach. In previous papers, in the homogeneous Poisson case, for the branching system in "low" dimensions, the limit was characterized by a long-range dependent Gaussian process called sub-fractional Brownian motion (sub-fBm), and this effect was attributed to the branching because it had appeared only in that case. An unexpected finding in this paper is that sub-fBm is more prevalent than previously thought. Namely, it is a natural ingredient of the limit process in the non-branching case (for "low" dimension), as well. On the other hand, fractional Brownian motion is not only related to systems in equilibrium (e.g., non-branching system with initial homogeneous Poisson measure), but it also appears here for a wider class of initial measures of quasi-homogeneous type.

http://arxiv.org/abs/1002.4152

9693. Bifractional Brownian motion with parameter $K\in(1,2)$

Author(s): Xavier Bardina and Khalifa Es-Sebaiy (SAMM)

Abstract: In this paper we introduce and study a self-similar Gaussian process that is the bifractional Brownian motion $B^{H,K}$ with parameters $H\in(0,1)$ and $K\in(1,2)$ such that $HK\in(0,1)$. A remarkable difference between the case $K\in(0,1)$ and our situation is that this process is a semimartingale when $2HK=1$.

http://arxiv.org/abs/1002.3680

9694. Scaling Limits for Random Quadrangulations of Positive Genus

Author(s): J\'er\'emie Bettinelli (LM-Orsay)

Abstract: We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given $g$, we consider, for every $n \ge 1$, a random quadrangulation $\q_n$ uniformly distributed over the set of all rooted bipartite quadrangulations of genus $g$ with $n$ faces. We view it as a metric space by endowing its set of vertices with the graph distance. We show that, as $n$ tends to infinity, this metric space, with distances rescaled by the factor $n^{-1/4}$, converges in distribution, at least along some subsequence, toward a limiting random metric space. This convergence holds in the sense of the Gromov-Hausdorff topology on compact metric spaces. We show that, regardless of the choice of the subsequence, the Hausdorff dimension of the limiting space is almost surely equal to 4. Our main tool is a bijection introduced by Chapuy, Marcus, and Schaeffer between the quadrangulations we consider and objects they call well-labeled $g$-trees. An important part of our study consists in determining the scaling limits of the latter.

http://arxiv.org/abs/1002.3682

9695. First-passage percolation with exponential times on a ladder

Author(s): Henrik Renlund

Abstract: We consider first-passage percolation on a ladder, i.e. the graph {0,1,...}*{0,1} where nodes at distance 1 are joined by an edge, and the times are exponentially i.i.d. with mean 1. We find an appropriate Markov chain to calculate an explicit expression for the time constant whose numerical value is approximately 0.6827. This time constant is the long-term average inverse speed of the process. We also calculate the average residual time.

http://arxiv.org/abs/1002.3709

9696. Generalized Polya urns via stochastic approximation

Author(s): Henrik Renlund

Abstract: We collect, survey and develop methods of (one-dimensional) stochastic approximation in a framework that seems suitable to handle fairly broad generalizations of Polya urns. To show the applicability of the results we determine the limiting fraction of balls in an urn with balls of two colors. We consider two models generalizing the Polya urn, in the first one ball is drawn and replaced with balls of (possibly) both colors according to which color was drawn. In the second, two balls are drawn simultaneously and replaced along with balls of (possibly) both colors according to what combination of colors were drawn.

http://arxiv.org/abs/1002.3716

9697. Singular perturbations to semilinear stochastic heat equations

Author(s): Martin Hairer

Abstract: We consider a class of singular perturbations to the stochastic heat equation or semilinear variations thereof. The interesting feature of these perturbations is that, as the small parameter epsilon tends to zero, their solutions converge to the 'wrong' limit, i.e. they do not converge to the solution obtained by simply setting epsilon = 0. A similar effect is also observed for some (formally) small stochastic perturbations of a deterministic semilinear parabolic PDE. Our proofs are based on a detailed analysis of the spatially rough component of the equations, combined with a judicious use of Gaussian concentration inequalities.

http://arxiv.org/abs/1002.3722

9698. Optimal stopping, Appell polynomials and Wiener-Hopf factorization representations of excessive functions of L\'evy processes

Author(s): Paavo Salminen

Abstract: In this paper we study the optimal stopping problem for L\'evy processes studied by Novikov and Shiryayev, Stochastics, 2007 In particular, we are interested in finding the representing measure of the value function. It is seen that that this can be expressed in terms of the Appell polynomials. An important tool in our approach and computations is the Wiener-Hopf factorization.

http://arxiv.org/abs/1002.3746

9699. Non-equilibrium dynamics of stochastic point processes: a dead-time approach

Author(s): Moritz Deger and Moritz Helias and Stefano Cardanobile and Fatihcan M. Atay and Stefan Rotter

Abstract: The Poisson process with dead time (PPD) is a widely used model for time series of events. Here we analyze non-equilibrium properties of an ensemble of PPDs. We derive a delay differential equation that describes the dynamics of the state of the ensemble. Analytical solutions are obtained for the time-dependent ensemble output rate in response to a step input. We also derive the mapping of periodic input to steady-state output, which we solve specifically for sinusoidal inputs. We are able to generalize the dynamics of the PPD to the case of random dead times, by which the method becomes applicable to a much larger class of stochastic point processes. Transient properties of the PPD are a recurring theme in many quantitative sciences, since a dead time after event detection is a feature of most technical counting devices. Our results are also relevant for the neurosciences because refractoriness is a characteristic of trains of action potentials emitted by nerve cells.

http://arxiv.org/abs/1002.3798

9700. On the unimodality of inverse positive stable laws

Author(s): Thomas Simon (LPP)

Abstract: We observe that the function $F_\alpha (x) = (1+ \alpha x^\alpha)e^{-x^\alpha}$ is completely monotone iff $\alpha \le \alpha_0$ for some $\alpha_0 \in ]2/3, 3/4[.$ This property is equivalent to the unimodality of the inverse positive $\alpha$-stable law. The random variable associated with $F_\alpha$ appears then in two different factorizations of the positive $\alpha$-stable distribution. Furthermore, it is infinitely divisible iff $\alpha \le \alpha_1$ for some $\alpha_1 \in ]2/3, \alpha_0[$ and self-decomposable iff $\alpha \le \alpha_2$ for some $\alpha_2 \in ]2/3, \alpha_1[.$

http://arxiv.org/abs/1002.3813

9701. Risk assessment for uncertain cash flows: Model ambiguity, discounting ambiguity, and the role of bubbles

Author(s): Beatrice Acciaio and Hans Foellmer and Irina Penner

Abstract: We study the risk assessment of uncertain cash flows in terms of dynamic convex risk measures for processes as introduced in Cheridito, Delbaen, and Kupper (2006). These risk measures take into account not only the amounts but also the timing of a cash flow. We discuss their robust representation in terms of suitably penalized probability measures on the optional sigma-field. This yields an explicit analysis both of model and discounting ambiguity. We focus on supermartingale criteria for different notions of time consistency. In particular we show how bubbles may appear in the dynamic penalization, and how they cause a breakdown of asymptotic safety of the risk assessment procedure.

http://arxiv.org/abs/1002.3627

9702. Interpolation and $\Phi$-moment inequalities of noncommutative martingales

Author(s): Turdebek N. Bekjan and Zeqian Chen

Abstract: This paper is devoted to the study of $\Phi$-moment inequalities for noncommutative martingales. In particular, we prove the noncommutative $\Phi$-moment analogues of martingale transformations, Stein's inequalities, Khintchine's inequalities for Rademacher's random variables, and Burkholder-Gundy's inequalities. The key ingredient is a noncommutative version of Marcinkiewicz type interpolation theorem for Orlicz spaces which we establish in this paper.

http://arxiv.org/abs/1002.3670

9703. Optimal investment with bounded VaR for power utility functions

Author(s): Behamar Chouaf and Serguei Pergamenchtchikov (LMRS)

Abstract: We consider the optimal investment problem for Black-Scholes type financial market with bounded VaR measure on the whole investment interval $[0,T]$. The explicit form for the optimal strategies is found.

http://arxiv.org/abs/1002.3681

9704. Dynamic risk measures

Author(s): Beatrice Acciaio and Irina Penner

Abstract: This paper gives an overview of the theory of dynamic convex risk measures for random variables in discrete time setting. We summarize robust representation results of conditional convex risk measures, and we characterize various time consistency properties of dynamic risk measures in terms of acceptance sets, penalty functions, and by supermartingale properties of risk processes and penalty functions.

http://arxiv.org/abs/1002.3794

9705. Energy efficiency of consecutive fragmentation processes

Author(s): Joaquin Fontbona and Nathalie Krell (IRMAR) and Servet Martinez

Abstract: We present a ?rst study on the energy required to reduce a unit mass fragment by consecutively using several devices, as it happens in the mining industry. Two devices are considered, which we represent as different stochastic fragmentation processes. Following the self-similar energy model introduced by Bertoin and Martinez, we compute the average energy required to attain a size x with this two-device procedure. We then asymptotically compare, as x goes to 0 or 1, its energy requirement with that of individual fragmentation processes. In particular, we show that for certain range of parameters of the fragmentation processes and of their energy cost-functions, the consecutive use of two devices can be asymptotically more efficient than using each of them separately, or conversely.

http://arxiv.org/abs/1002.3460

9706. A short proof of Cramer's theorem in R

Author(s): Raphael Cerf and Pierre Petit

Abstract: We expose here a short proof of Cramer's theorem in R based on convex duality.

http://arxiv.org/abs/1002.3496

9707. Asymptotic risks of Viterbi segmentation

Author(s): Kristi Kuljus and J\"uri Lember

Abstract: We consider the maximum likelihood (Viterbi) alignment of a hidden Markov model (HMM). In an HMM, the underlying Markov chain is usually hidden and the Viterbi alignment is often used as the estimate of it. This approach will be referred to as the Viterbi segmentation. The goodness of the Viterbi segmentation can be measured by several risks. In this paper, we prove the existence of asymptotic risks. Being independent of data, the asymptotic risks can be considered as the characteristics of the model that illustrate the long-run behavior of the Viterbi segmentation.

http://arxiv.org/abs/1002.3509

9708. Multi-type TASEP in discrete time

Author(s): James Martin and Philipp Schmidt

Abstract: The TASEP (totally asymmetric simple exclusion process) is a basic model for an one-dimensional interacting particle system with non-reversible dynamics. Despite the simplicity of the model it shows a very rich and interesting behaviour. In this paper we study some aspects of the TASEP in discrete time and compare the results to the recently obtained results for the TASEP in continuous time. In particular we focus on stationary distributions for multi-type models, speeds of second-class particles, collision probabilities and the "speed process". In discrete time, jump attempts may occur at different sites simultaneously, and the order in which these attempts are processed is important; we consider various natural update rules.

http://arxiv.org/abs/1002.3539

9709. Approximation by Log-Concave Distributions with Applications to Regression

Author(s): Lutz Duembgen and Richard Samworth and Dominic Schuhmacher

Abstract: We study the approximation of arbitrary distributions P on d-dimensional space by distributions with log-concave density. Approximation means minimizing a Kullback-Leibler type functional. We show that such an approximation exists if, and only if, P has finite first moments and is not concentrated on some hyperplane. Furthermore we show that this approximation depends continuously on P with respect to Mallows' distance D_1. This result implies consistency of the maximum likelihood estimator of a log-concave density under fairly general conditions. It also allows us to prove existence and consistency of estimators in regression models with a response Y = m(X) + E, where X and E are independent, m(.) belongs to a certain class of regression functions while E is a random error with log-concave density.

http://arxiv.org/abs/1002.3448

9710. Double scaling limits of random matrices and minimal (2m,1) models: the merging of two cuts in a degenerate case

Author(s): Olivier Marchal and Mattia Cafasso

Abstract: In this article, we show that the double scaling limit correlation functions of a random matrix model when two cuts merge with degeneracy $2m$ (i.e. when $y\sim x^{2m}$ for arbitrary values of the integer $m$) are the same as the determinental formulae defined by conformal $(2m,1)$ models. Our approach follows the one developped by Berg\`{e}re and Eynard in \cite{BergereEynard} and uses a Lax pair representation of the conformal $(2m,1)$ models (giving Painleve II integrable hierarchy) as suggested by Bleher and Eynard in \cite{BleherEynard}. In particular we define Baker-Akhiezer functions associated to the Lax pair to construct a kernel which is then used to compute determinental formulae giving the correlation functions of the double scaling limit of a matrix model near the merging of two cuts.

http://arxiv.org/abs/1002.3347

9711. On monotone convolution and monotone infinite divisivility

Author(s): Takahiro Hasebe

Abstract: This article is focused on properties of monotone convolutions. A criterion for infinite divisibility and time evolution of convolution semigroups are mainly studied. In particular, we clarify that many analogues of the classical results of L\'{e}vy processes hold such as characterizations of subordinators and strictly stable distributions.

http://arxiv.org/abs/1002.3430

9712. Rumor Spreading on Random Regular Graphs and Expanders

Author(s): Nikolaos Fountoulakis and Konstantinos Panagiotou

Abstract: Broadcasting algorithms are important building blocks of distributed systems. In this work we investigate the typical performance of the classical and well-studied push model. Assume that initially one node in a given network holds some piece of information. In each round, every one of the informed nodes chooses independently a neighbor uniformly at random and transmits the message to it. In this paper we consider random networks where each vertex has degree d, which is at least 3, i.e., the underlying graph is drawn uniformly at random from the set of all d-regular graphs with n vertices. We show that with probability 1 - o(1) the push model broadcasts the message to all nodes within (1 + o(1))C_d ln n rounds, where C_d = 1/ ln(2(1-1/d)) - 1/(d ln(1 - 1/d)). In particular, we can characterize precisely the effect of the node degree to the typical broadcast time of the push model. Moreover, we consider pseudo-random regular networks, where we assume that the degree of each node is very large. There we show that the broadcast time is (1+o(1))C ln n with probability 1 - o(1), where C= 1/ ln 2 + 1, is the limit of C_d as d grows.

http://arxiv.org/abs/1002.3518

9713. Uniform Large deviations for infinite dimensional stochastic systems with jumps

Author(s): Vasileios Maroulas

Abstract: Uniform large deviation principles for positive functionals of all equivalent types of infinite dimensional Brownian motions acting together with a Poisson random measure are established. The core of our approach is a variational representation formula which for an infinite sequence of i.i.d real Brownian motions and a Poisson random measure was shown in [5].

http://arxiv.org/abs/1002.3290

9714. Coarse-grained modeling of multiscale diffusions: the p-variation estimates

Author(s): Anastasia Papavasiliou

Abstract: We study the problem of estimating parameters of the limiting equation of a multiscale diffusion in the case of averaging and homogenization, given data from the corresponding multiscale system. First, we review some recent results that make use of the maximum likelihood of the limiting equation. In particular, it has been shown that in the averaging case, the MLE will be asymptotically consistent in the limit while in the homogenization case, the MLE will be asymptotically consistent only if we subsample the data. Then, we focus on the problem of estimating the diffusion coefficient. We suggest a novel approach that makes use of the total $p$-variation, as defined in the theory of rough paths and avoids the subsampling step. The method is applied to a multiscale OU process.

http://arxiv.org/abs/1002.3241

9715. A discrete time neural network model with spiking neurons II. Dynamics with noise

Author(s): B. Cessac

Abstract: We provide rigorous and exact results characterizing the statistics of spike trains in a network of leaky integrate and fire neurons, where time is discrete and where neurons are submitted to noise, without restriction on the synaptic weights. We show the existence and uniqueness of an invariant measure of Gibbs type and discuss its properties. We also discuss Markovian approximations and relate them to the approaches currently used in computational neuroscience to analyse experimental spike trains statistics.

http://arxiv.org/abs/1002.3275

9716. Distribution of Relative Edge Density of the Underlying Graphs Based on a Random Digraph Family

Author(s): Elvan Ceyhan

Abstract: The data-random graphs called proximity catch digraphs (PCDs) have been introduced recently and have applications in pattern recognition and spatial pattern analysis. A PCD is a random directed graph (i.e., digraph) which is constructed from data using the relative positions of the points from various classes. Different PCDs result from different definitions of the proximity region associated with each data point. We consider the underlying graphs based on a family of PCDs which is determined by a family of parameterized proximity maps called proportional-edge proximity map. The graph invariant we investigate is the relative edge density of the underlying graphs. We demonstrate that, properly scaled, relative edge density of the underlying graphs is a U-statistic, and hence obtain the asymptotic normality of the relative edge density for data from any distribution that satisfies mild regulatory conditions. By detailed probabilistic and geometric calculations, we compute the explicit form of the asymptotic normal distribution for uniform data on a bounded region. We also compare the relative edge densities of the two types of the underlying graphs and the relative arc density of the PCDs. The approach presented here is also valid for data in higher dimensions.

http://arxiv.org/abs/1002.2957

9717. Khasminskii-Type Theorem and LaSalle-Type Theorem for Stochastic Evolution Delay Equations

Author(s): Jianhai Bao and Xuerong Mao and Chenggui Yuan

Abstract: In this paper we study the well-known Khasminskii-Type Theorem, i.e. the existence and uniqueness of solutions of stochastic evolution delay equations, under local Lipschitz condition, but without linear growth condition. We then establish one stochastic LaSalle-type theorem for asymptotic stability analysis of strong solutions. Moreover, several examples are established to illustrate the power of our theories.

http://arxiv.org/abs/1002.3116

9718. Subgraphs of dense random graphs with specified degrees

Author(s): Brendan D McKay

Abstract: Let d = (d1, d2, ..., dn) be a vector of non-negative integers with even sum. We prove some basic facts about the structure of a random graph with degree sequence d, including the probability of a given subgraph or induced subgraph. Although there are many results of this kind, they are restricted to the sparse case with only a few exceptions. Our focus is instead on the case where the average degree is approximately a constant fraction of n. Our approach is the multidimensional saddle-point method. This extends the enumerative work of McKay and Wormald (1990) and is analogous to the theory developed for bipartite graphs by Greenhill and McKay (arXiv:math/0701600, 2009).

http://arxiv.org/abs/1002.3018

9719. A Sharp Liouville Theorem for Elliptic Operators

Author(s): Enrico Priola and Feng-Yu Wang

Abstract: We introduce a new condition on elliptic operators $L= {1/2}\triangle + b \cdot \nabla $ which ensures the validity of the Liouville property for bounded solutions to $Lu=0$ on $\R^d$. Such condition is sharp when $d=1$. We extend our Liouville theorem to more general second order operators in non-divergence form assuming a Cordes type condition.

http://arxiv.org/abs/1002.3055

9720. Random Strict Partitions and Determinantal Point Processes

Author(s): Leonid Petrov

Abstract: In this note we present new examples of determinantal point processes with infinitely many particles. The particles live on the half-lattice {1,2,...} or on the open half-line (0,+\infty). The main result is the computation of the correlation kernels. They have integrable form and are expressed through the Euler gamma function (the lattice case) and the classical Whittaker functions (the continuous case). Our processes are obtained via a limit transition from a model of random strict partitions introduced by Borodin (1997) in connection with the problem of harmonic analysis for projective characters of the infinite symmetric group.

http://arxiv.org/abs/1002.2714

9721. Long Strange Segments, Ruin Probabilities and the Effect of Memory on Moving Average Processes

Author(s): Souvik Ghosh and Gennady Samorodnitsky

Abstract: We obtain the rate of growth of long strange segments and the rate of decay of infinite horizon ruin probabilities for a class of infinite moving average processes with exponentially light tails. The rates are computed explicitly. We show that the rates are very similar to those of an i.i.d. process as long as moving average coefficients decay fast enough. If they do not, then the rates are significantly different. This demonstrates the change in the length of memory in a moving average process associated with certain changes in the rate of decay of the coefficients.

http://arxiv.org/abs/1002.2751

9722. On the trace of branching random walk

Author(s): Itai Benjamini and Sebastian M\"uller

Abstract: We study branching random walk on Cayley graphs. A first result is that the trace of a transient branching random walk on a Cayley graph is a.s. transient for simple random walk. In addition, it has a.s. critical percolation probability less than one and exponential volume growth. The proofs rely on the fact that the trace induces an invariant percolation on the family tree of the branching random walk. Furthermore, we prove that the trace is a.s. strongly recurrent for any branching random walk. This follows from the observation that the trace, after appropriate biasing of the root, defines a unimodular measure. All the results hold more generally for branching random walk on unimodular random graphs.

http://arxiv.org/abs/1002.2781

9723. Competing particle systems evolving by interacting Levy processes

Author(s): Mykhaylo Shkolnikov

Abstract: We consider finite and infinite systems of particles on the real line and half-line evolving in continuous time. Hereby, the particles are driven by i.i.d. Levy processes endowed with rank-dependent drift and diffusion coefficients. In the finite systems we show that the processes of gaps in the respective particle configurations possess unique invariant distributions and prove the convergence of the gap processes to the latter in the total variation distance, assuming a bound on the jumps of the Levy processes. In the infinite case we show that the gap process of the particle system on the half-line is tight for appropriate initial conditions and same drift and diffusion coefficients for all particles. Applications of such processes include the modelling of capital distributions among the ranked participants in a financial market, the stability of certain stochastic queueing and storage networks and the study of the Sherrington-Kirkpatrick model of spin glasses.

http://arxiv.org/abs/1002.2811

9724. Analysis on Path Spaces over Riemmannian Manifolds with Boundary

Author(s): Feng-Yu Wang

Abstract: By using Hsu's multiplicative functional for the Neumann heat equation, a natural damped gradient operator is defined for the reflecting Brownian motion on compact manifolds with boundary. This operator is linked to quasi-invariant flows in terms of a integration by parts formula, which leads to the standard log-Sobolev inequality for the associated Dirichlet form on the path space.

http://arxiv.org/abs/1002.2887

9725. Coupling and Strong Feller Property for Ornstein-Uhlenbeck Jump Processes

Author(s): Feng-Yu Wang

Abstract: Coupling and strong Feller property are investigated for the linear SDE on $\R^d$: $$\d X_t= A X_t\d t+ \d L_t,$$ where $A$ is a $d\times d$ real matrix and $L_t$ is a L\'evy process with L\'evy measure $\nu$ on $\R^d$. Assume that $\nu(\d z)\ge \rr_0(z)\d z$ for some $\rr_0\ge 0$. If $A \le 0$ and $\int_{B(x_0,\vv)} \rr_0(z)^{-1}\d z<\infty$ holds for some $x_0\in \R^d$ and some $\vv>0$, then the associated Markov transition probability $P_t(x,\d y)$ satisfies $$\|P_t (x, \cdot)- P_t (y, \cdot)\|_{var} \le \ff{C(1+|x-y|)}{\ss t}, x,y\in \R^d, t>0$$ for some constant $C>0$, which is sharp for large $t$ and implies that the process has successful couplings. If $\rr_0\in C(\R^d\setminus \{0\})$ with $\int_{\R^d}\rr_0(z)\d z=\infty$, then the process is strong Feller.

http://arxiv.org/abs/1002.2890

9726. Distance statistics in quadrangulations with no multiple edges and the geometry of minbus

Author(s): J. Bouttier and E. Guitter

Abstract: We present a detailed calculation of the distance-dependent two-point function for quadrangulations with no multiple edges. Various discrete observables measuring this two-point function are computed and analyzed in the limit of large maps. For large distances and in the scaling regime, we recover the same universal scaling function as for general quadrangulations. We then explore the geometry of "minimal neck baby universes" (minbus), which are the outgrowths to be removed from a general quadrangulation to transform it into a quadrangulation with no multiple edges, the "mother universe". We give a number of distance-dependent characterizations of minbus, such as the two-point function inside a minbu or the law for the distance from a random point to the mother universe.

http://arxiv.org/abs/1002.2552

9727. Characteristic functions of affine processes via calculus of their operator symbols

Author(s): Joerg Kampen

Abstract: The characteristic functions of multivariate Feller processes with generator of affine type, and with smooth symbol functions have an explicit representation in terms of power series with rational number coefficients and with monmoms consisting of powers of the the symbol functions and formal derivatives of the symbol functions. The power series repesentations are convergent globally in time and on bounded domains of arbitrary size. Generalized symbol functions can be derived leading to power series expansions which are convergent on arbitrary domains in special cases. The rational number coefficients can be efficiently computed by an integer recursion. As a numerical consequence characteristic functions of multivariate affine processes can be efficiently computed from the symbol function avoiding computation of the generalized Riccati equations (an observation first made recently in a more general context).

http://arxiv.org/abs/1002.2764

9728. Optimal consumption and investment with bounded downside risk for power utility functions

Author(s): Claudia Kluppelberg and Serguei Pergamenchtchikov (LMRS)

Abstract: We investigate optimal consumption and investment problems for a Black-Scholes market under uniform restrictions on Value-at-Risk and Expected Shortfall. We formulate various utility maximization problems, which can be solved explicitly. We compare the optimal solutions in form of optimal value, optimal control and optimal wealth to analogous problems under additional uniform risk bounds. Our proofs are partly based on solutions to Hamilton-Jacobi-Bellman equations, and we prove a corresponding verification theorem. This work was supported by the European Science Foundation through the AMaMeF programme.

http://arxiv.org/abs/1002.2487

9729. Functional Ito calculus and stochastic integral representation of martingales

Author(s): Rama Cont and David-Antoine Fournie

Abstract: We develop a non-anticipative calculus for functionals of a continuous semimartingale, using a notion of pathwise functional derivative. A functional extension of the Ito formula is derived and used to obtain a constructive martingale representation theorem for a class of continuous martingales verifying a regularity property. By contrast with the Clark-Haussmann-Ocone formula, this representation involves non-anticipative quantities which can be computed pathwise. These results are used to construct a weak derivative acting on square-integrable martingales, which is shown to be the inverse of the Ito integral, and derive an integration by parts formula for Ito stochastic integrals. We show that this weak derivative may be viewed as a non-anticipative "lifting" of the Malliavin derivative. Regular functionals of an Ito martingale which have the local martingale property are characterized as solutions of a functional differential equation, for which a uniqueness result is given.

http://arxiv.org/abs/1002.2446

9730. Plaquettes, Spheres, and Entanglement

Author(s): Geoffrey R. Grimmett and Alexander E. Holroyd

Abstract: The high-density plaquette percolation model in d dimensions contains a surface that is homeomorphic to the (d-1)-sphere and encloses the origin. This is proved by a path-counting argument in a dual model. When d=3, this permits an improved lower bound on the critical point p_e of entanglement percolation, namely p_e >= \mu^-2 where \mu is the connective constant for self-avoiding walks on Z^3. Furthermore, when the edge density p is below this bound, the radius of the entanglement cluster containing the origin has an exponentially decaying tail.

http://arxiv.org/abs/1002.2623

9731. Optimal consumption and investment with bounded downside risk measures for logarithmic utility functions

Author(s): Claudia Kluppelberg and Serguei Pergamenchtchikov (LMRS)

Abstract: We investigate optimal consumption problems for a Black-Scholes market under uniform restrictions on Value-at-Risk and Expected Shortfall for logarithmic utility functions. We find the solutions in terms of a dynamic strategy in explicit form, which can be compared and interpreted. This paper continues our previous work, where we solved similar problems for power utility functions.

http://arxiv.org/abs/1002.2486

9732. Optimal consumption and investment with bounded downside risk for power utility functions

Author(s): Claudia Kluppelberg and Serguei Pergamenchtchikov (LMRS)

Abstract: We investigate optimal consumption and investment problems for a Black-Scholes market under uniform restrictions on Value-at-Risk and Expected Shortfall. We formulate various utility maximization problems, which can be solved explicitly. We compare the optimal solutions in form of optimal value, optimal control and optimal wealth to analogous problems under additional uniform risk bounds. Our proofs are partly based on solutions to Hamilton-Jacobi-Bellman equations, and we prove a corresponding verification theorem. This work was supported by the European Science Foundation through the AMaMeF programme.

http://arxiv.org/abs/1002.2487

9733. Geometric ergodicity for families of homogeneous Markov chains

Author(s): Leonid Galtchouk (IRMA) and Serguei Pergamenchtchikov (LMRS)

Abstract: In this paper we find nonasymptotic exponential upper bounds for the deviation in the ergodic theorem for families of homogeneous Markov processes. We find some sufficient conditions for geometric ergodicity uniformly over a parametric family. We apply this property to the nonasymptotic nonparametric estimation problem for ergodic diffusion processes.

http://arxiv.org/abs/1002.2341

9734. Duality theory for Markov processes: Part 1

Author(s): Ronald Getoor

Abstract: This is the first part of a possible monograph on the duality of Markov processes. It contains a proof of Fitzsimmons' existence theorem of a moderate Markov dual process relative to an excessive measure, m, together with the necessary preliminary material. Then this is applied to prove the correspondence between optional copredictable homogenous random measures and sigma finite measures not charging m-exceptional sets again following Fitzsimmons. The second part which may never be written would deal with duality proper including results from, but not limited to, my joint paper with P. J. Fitzsimmons"Potential Theory of Moderate Markov Dual Processes" which appeared in Potential Anal.(2009) 31:275-310. Complete proofs of all results not appearing in standard books are given with the one exception of Dellacherie's result characterizing semipolar sets.

http://arxiv.org/abs/1002.2399

9735. Viability, Invariance and Reachability for Controlled Piecewise Deterministic Markov Processes Associated to Gene Networks

Author(s): D. Goreac

Abstract: We aim at characterizing viability, invariance and some reachability properties of controlled piecewise deterministic Markov processes (PDMPs). Using analytical methods from the theory of viscosity solutions, we establish criteria for viability and invariance in terms of the first order normal cone. We also investigate reachability of arbitrary open sets. The method is based on viscosity techniques and duality for some associated linearized problem. The theoretical results are applied to general On/Off systems, Cook's model for haploinssuficiency, and a stochastic model for bacteriophage lambda.

http://arxiv.org/abs/1002.2242

9736. Forward-convex convergence of sequences in $\mathbb{L}^0_+$

Author(s): Constantinos Kardaras and Gordan Zitkovic

Abstract: For a sequence in $\mathbb{L}^0_+$, we provide simple necessary and sufficient conditions to ensure that each sequence of its forward convex combinations converges to the same limit. These conditions correspond to a measure-free version of the notion of uniform integrability and are related to the numeraire problem of mathematical finance.

http://arxiv.org/abs/1002.1889

9737. Impulse control problem with switching technology

Author(s): Rim Amami

Abstract: We consider an impulse control problem in infinite horizon applied with switching technology. We suppose that the firm decides at certain moments (impulse moments) to switch technology, leading to a jump of the firm value. We show that the value function for such problems satisfies a dynamic programming principle version. Our objective is to look for an optimal strategy which maximizes the value function associated with a switching problem.

http://arxiv.org/abs/1002.2086

9738. Applying coupon-collecting theory to computer-aided assessments

Author(s): Charles M. Goldie (University of Sussex) and Rosie Cornish (University of Bristol), Carol L. Robinson (Loughborough University)

Abstract: Computer-based tests with randomly generated questions allow a large number of different tests to be generated. Given a fixed number of alternatives for each question, the number of tests that need to be generated before all possible questions have appeared is surprisingly low.

http://arxiv.org/abs/1002.2114

9739. The fractional Poisson measure in infinite dimensions

Author(s): Maria Joao Oliveira and Habib Ouerdiane and Jose Luis da Silva and R. Vilela Mendes

Abstract: The Mittag-Leffler function $E_{\alpha}$ being a natural generalization of the exponential function, an infinite-dimensional version of the fractional Poisson measure would have a characteristic functional \[ C_{\alpha}(\varphi) :=E_{\alpha}(\int (e^{i\varphi(x)}-1)d\mu (x)) \] which we prove to fulfill all requirements of the Bochner-Minlos theorem. The identity of the support of this new measure with the support of the infinite-dimensional Poisson measure ($\alpha =1$) allows the development of a fractional infinite-dimensional analysis modeled on Poisson analysis through the combinatorial harmonic analysis on configuration spaces. This setting provides, in particular, explicit formulas for annihilation, creation, and second quantization operators. In spite of the identity of the supports, the fractional Poisson measure displays some noticeable differences in relation to the Poisson measure, which may be physically quite significant.

http://arxiv.org/abs/1002.2124

9740. Bernstein type's concentration inequalities for symmetric Markov processes

Author(s): Fuqing Gao and Arnaud Guillin and Liming Wu

Abstract: Using the method of transportation-information inequality introduced in \cite{GLWY}, we establish Bernstein type's concentration inequalities for empirical means $\frac 1t \int_0^t g(X_s)ds$ where $g$ is a unbounded observable of the symmetric Markov process $(X_t)$. Three approaches are proposed : functional inequalities approach ; Lyapunov function method ; and an approach through the Lipschitzian norm of the solution to the Poisson equation. Several applications and examples are studied.

http://arxiv.org/abs/1002.2163

9741. The Ghirlanda-Guerra identities for mixed p-spin model

Author(s): Dmitry Panchenko

Abstract: We show that, under the conditions known to imply the validity of the Parisi formula, if the generic Sherrington-Kirkpatrick Hamiltonian contains a $p$-spin term then the Ghirlanda-Guerra identities for the $p$th power of the overlap hold in a strong sense without averaging. This implies strong version of the extended Ghirlanda-Guerra identities for mixed $p$-spin models than contain terms for all even $p\geq 2$ and $p=1.$

http://arxiv.org/abs/1002.2190

9742. On the strict comparison theorem for $G$-expectations

Author(s): Xinpeng Li

Abstract: In this paper, we prove two forms of strict comparison theorem for $X,Y\in L_G^1(\Omega)$. Furthermore, if $X,Y\in Lip(\Omega)$ and $\x>0$, we give a necessary and sufficient condition under which the strict comparison theorem holds.

http://arxiv.org/abs/1002.1765

9743. On isoperimetric sets of radially symmetric measures

Author(s): Alexander V. Kolesnikov and Roman I. Zhdanov

Abstract: We study the isoperimetric problem for the radially symmetric measures. Applying the spherical symmetrization procedure and variational arguments we reduce this problem to a one-dimensional ODE of the second order. Solving numerically this ODE we get an empirical description of isoperimetric regions of the planar radially symmetric exponential power laws. We also prove some isoperimetric inequalities for the log-convex measures. We show, in particular, that the symmetric balls of large size are isoperimetric sets for strictly log-convex and radially symmetric measures. We also establish a comparison theorem for the products of the one-dimensional log-convex measures.

http://arxiv.org/abs/1002.1829

9744. A local limit theorem for random walks in random scenery and on randomly oriented lattices

Author(s): Fabienne Castell (LATP) and Nadine Guillotin-Plantard (UCB and ICJ) and Fran\c{c}oise P\`ene (LM), Bruno Schapira (LM-Orsay)

Abstract: Random walks in random scenery are processes defined by $Z_n:=\sum_{k=1}^n\xi_{X_1+...+X_k}$, where $(X_k,k\ge 1)$ and $(\xi_y,y\in\mathbb Z)$ are two independent sequences of i.i.d. random variables. We assume here that their distributions belong to the normal domain of attraction of stable laws with index $\alpha\in (0,2]$ and $\beta\in (0,2]$ respectively. These processes were first studied by H. Kesten and F. Spitzer, who proved the convergence in distribution when $\alpha\neq 1$ and as $n\to \infty$, of $n^{-\delta}Z_n$, for some suitable $\delta>0$ depending on $\alpha$ and $\beta$. Here we are interested in the convergence, as $n\to \infty$, of $n^\delta{\mathbb P}(Z_n=\lfloor n^{\delta} x\rfloor)$, when $x\in \RR$ is fixed. We also consider the case of random walks on randomly oriented lattices for which we obtain similar results.

http://arxiv.org/abs/1002.1878

9745. Percolation in invariant Poisson graphs with i.i.d. degrees

Author(s): Maria Deijfen and Olle Haggstrom and and Alexander E. Holroyd

Abstract: Let each point of a homogeneous Poisson process in R^d independently be equipped with a random number of stubs (half-edges) according to a given probability distribution mu on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution mu. Leaving aside degenerate cases, we prove that for any mu there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme that is a natural extension of Gale-Shapley stable marriage, we give sufficient conditions on mu for the absence and presence of infinite components.

http://arxiv.org/abs/1002.1943

9746. Stochastic Analysis of Non-slotted Aloha in Wireless Ad-Hoc Networks

Author(s): Bartek Blaszczyszyn (INRIA Rocquencourt) and Paul Muhlethaler (INRIA Rocquencourt)

Abstract: In this paper we propose two analytically tractable stochastic models of non-slotted Aloha for Mobile Ad-hoc NETworks (MANETs): one model assumes a static pattern of nodes while the other assumes that the pattern of nodes varies over time. Both models feature transmitters randomly located in the Euclidean plane, according to a Poisson point process with the receivers randomly located at a fixed distance from the emitters. We concentrate on the so-called outage scenario, where a successful transmission requires a Signal-to-Interference-and-Noise Ratio (SINR) larger than a given threshold. With Rayleigh fading and the SINR averaged over the duration of the packet transmission, both models lead to closed form expressions for the probability of successful transmission. We show an excellent matching of these results with simulations. Using our models we compare the performances of non-slotted Aloha to previously studied slotted Aloha. We observe that when the path loss is not very strong both models, when appropriately optimized, exhibit similar performance. For stronger path loss non-slotted Aloha performs worse than slotted Aloha, however when the path loss exponent is equal to 4 its density of successfully received packets is still 75% of that in the slotted scheme. This is still much more than the 50% predicted by the well-known analysis where simultaneous transmissions are never successful. Moreover, in any path loss scenario, both schemes exhibit the same energy efficiency.

http://arxiv.org/abs/1002.1629

9747. Increasing the chromatic number of a random graph

Author(s): N. Alon and B. Sudakov

Abstract: What is the minimum number of edges that have to be added to the random graph $G=G_{n,0.5}$ in order to increase its chromatic number $\chi=\chi(G)$ by one percent ? One possibility is to add all missing edges on a set of $1.01 \chi$ vertices, thus creating a clique of chromatic number $1.01 \chi$. This requires, with high probability, the addition of $\Omega(n^2/\log^2 n)$ edges. We show that this is tight up to a constant factor, consider the question for more general random graphs $G_{n,p}$ with $p=p(n)$, and study a local version of the question as well. The question is motivated by the study of the resilience of graph properties, initiated by the second author and Vu, and improves one of their results.

http://arxiv.org/abs/1002.1748

9748. How to lose as little as possible

Author(s): Vittorio Addona and Stan Wagon and and Herb Wilf

Abstract: Suppose Alice has a coin with heads probability $q$ and Bob has one with heads probability $p>q$. Now each of them will toss their coin $n$ times, and Alice will win iff she gets more heads than Bob does. Evidently the game favors Bob, but for the given $p,q$, what is the choice of $n$ that maximizes Alice's chances of winning? The problem of determining the optimal $N$ first appeared in \cite{wa}. We show that there is an essentially unique value $N(q,p)$ of $n$ that maximizes the probability $f(n)$ that the weak coin will win, and it satisfies $\frac{1}{2(p-q)}-\frac12\le N(q,p)\le \frac{\max{(1-p,q)}}{p-q}$. The analysis uses the multivariate form of Zeilberger's algorithm to find an indicator function $J_n(q,p)$ such that $J>0$ iff $n

http://arxiv.org/abs/1002.1763

9749. Forward-convex convergence of sequences in $\mathbb{L}^0_+$

Author(s): Constantinos Kardaras and Gordan Zitkovic

Abstract: For a sequence in $\mathbb{L}^0_+$, we provide simple necessary and sufficient conditions to ensure that each sequence of its forward convex combinations converges to the same limit. These conditions correspond to a measure-free version of the notion of uniform integrability and are related to the numeraire problem of mathematical finance.

http://arxiv.org/abs/1002.1889

9750. The Bismut-Elworthy-Li type formulae for stochastic differential equations with jumps

Author(s): Atsushi Takeuchi

Abstract: Consider jump-type stochastic differential equations with the drift, diffusion and jump terms. Logarithmic derivatives of densities for the solution process are studied, and the Bismut-Elworthy-Li type formulae can be obtained under the uniformly elliptic condition on the coefficients of the diffusion and jump terms. Our approach is based upon the Kolmogorov backward equation by making full use of the Markovian property of the process.

http://arxiv.org/abs/1002.1384

9751. Stochastic Power Law Fluids:the Existence and the Uniqueness of the Weak Solution

Author(s): Nobuo Yoshida and Yutaka Terasawa

Abstract: We consider a SPDE (stochastic partial differential equation) which describes the velocity field of a viscous, incompressible non-Newtonian fluid subject to a random perturbation. Here, the extra stress tensor of the fluid is given by a polynomial of degree $p-1$ of the deformation rate tensor, while the colored noise is considered as the random perturbation. We investigate the existence and the uniqueness of the weak solution to this SPDE.

http://arxiv.org/abs/1002.1431

9752. Extension of the Olkin and Rubin Characterization to the Wishart distribution on homogeneous cones

Author(s): Imen Boutouria and Abdelhamid Hassairi and Helene Massam

Abstract: The Wishart distribution on an homogeneous cone is a generalization of the Riesz distribution on a symmetric cone which corresponds to a given graph. The paper extends to this distribution, the famous Olkin and Rubin characterization of the ordinary Wishart distribution on symmetric matrices.

http://arxiv.org/abs/1002.1451

9753. Hitting densities for spectrally positive stable processes

Author(s): Thomas Simon (LPP)

Abstract: A multiplicative identity in law connecting the hitting times of completely asymmetric $\alpha-$stable L\'evy processes in duality is established. In the spectrally positive case, this identity allows with an elementary argument to compute fractional moments and to get series representations for the density. We also prove that the hitting times are unimodal as soon as $\alpha\le 3/2.$ Analogous results are obtained, in a much simplified manner, for the first passage time across a positive level.

http://arxiv.org/abs/1002.1540

9754. Generalized Dirichlet distributions on the ball and moments

Author(s): Franck Barthe and Fabrice Gamboa and Li-Vang Lozada-Chang and Alain Rouault

Abstract: The geometry of unit $N$-dimensional $\ell_{p}$ balls has been intensively investigated in the past decades. A particular topic of interest has been the study of the asymptotics of their projections. Apart from their intrinsic interest, such questions have applications in several probabilistic and geometric contexts (Barthe et al. 2005). In this paper, our aim is to revisit some known results of this flavour with a new point of view. Roughly speaking, we will endow the ball with some kind of Dirichlet distribution that generalizes the uniform one and will follow the method developed in Skibinsky (1967), Chang et al. (1993) in the context of the randomized moment space. The main idea is to build a suitable coordinate change involving independent random variables. Moreover, we will shed light on a nice connection between the randomized balls and the randomized moment space.

http://arxiv.org/abs/1002.1544

9755. Space-time correlations of a Gaussian interface

Author(s): Francois M. Dunlop

Abstract: The serial harness introduced by Hammersley is equivalent, in the Gaussian case, to the Gaussian Solid-On-Solid interface model with parallel heat bath dynamics. Here we consider sub-lattice parallel dynamics, and give exact results about relaxation dynamics, based on the equivalence to the infinite time limit of a time periodic random field. We also give a numerical comparison to the harness process in continuous time studied by Hsiao and by Ferrari, Niederhauser and Pechersky.

http://arxiv.org/abs/1002.1604

9756. Adaptive LASSO-type estimation for ergodic diffusion processes

Author(s): A. De Gregorio and S.M. Iacus

Abstract: The LASSO is a widely used statistical methodology for simultaneous estimation and variable selection. In the last years, many authors analyzed this technique from a theoretical and applied point of view. We introduce and study the adaptive LASSO problem for discretely observed ergodic diffusion processes. We prove oracle properties also deriving the asymptotic distribution of the LASSO estimator. Our theoretical framework is based on the random field approach and it applied to more general families of regular statistical experiments in the sense of Ibragimov-Hasminskii (1981). Furthermore, we perform a simulation and real data analysis to provide some evidence on the applicability of this method.

http://arxiv.org/abs/1002.1312

9757. Probabilistic interpretation of the M\"obius function identity and the Riemann Hypothesis

Author(s): R. M. Abrarov and R. M. Abrarov

Abstract: We obtained the probabilities for the values of the M\"obius function for arbitrary numbers and found that the asymptotic densities of the squarefree integers among the odd and even numbers are $8/\pi^2$ and $4/\pi^2$, respectively. It is determined that statistics of successive outcomes of the M\"obius function for very large squarefree odd and even numbers behaves similar to statistics of heads and tails of two flipping coins. These preliminary results are giving arguments supporting the Riemann Hypothesis. Its plausibility is based on statistical phenomena for integers.

http://arxiv.org/abs/1002.1682

9758. Heat Kernel Estimate for $\Delta+\Delta^{\alpha/2}$ in $C^{1,1}$ open sets

Author(s): Zhen-Qing Chen and Panki Kim and Renming Song

Abstract: We consider a family of pseudo differential operators $\{\Delta+ a^\alpha \Delta^{\alpha/2}; a\in (0, 1]\}$ on $\bR^d$ for every $d\geq 1$ that evolves continuously from $\Delta$ to $\Delta + \Delta^{\alpha/2}$, where $\alpha \in (0, 2)$. It gives rise to a family of L\'evy processes $\{X^a, a\in (0, 1]\}$ in $\bR^d$, where $X^a$ is the sum of a Brownian motion and an independent symmetric $\alpha$-stable process with weight $a$. We establish sharp two-sided estimates for the heat kernel of $\Delta + a^{\alpha} \Delta^{\alpha/2}$ with zero exterior condition in a family of open subsets, including bounded $C^{1, 1}$ (possibly disconnected) open sets. This heat kernel is also the transition density of the sum of a Brownian motion and an independent symmetric $\alpha$-stable process with weight $a$ in such open sets. Our result is the first sharp two-sided estimates for the transition density of a Markov process with both diffusion and jump components in open sets. Moreover, our result is uniform in $a$ in the sense that the constants in the estimates are independent of $a\in (0, 1]$ so that it recovers the Dirichlet heat kernel estimates for Brownian motion by taking $a\to 0$. Integrating the heat kernel estimates in time $t$, we recover the two-sided sharp uniform Green function estimates of $X^a$ in bounded $C^{1,1}$ open sets in $\bR^d$, which were recently established in \cite{CKSV2} by using a completely different approach.

http://arxiv.org/abs/1002.1121

9759. Random sampling of lattice paths with constraints, via transportation

Author(s): Lucas Gerin (MODAL'x)

Abstract: We discuss a Monte Carlo Markov Chain (MCMC) procedure for the random sampling of some one-dimensional lattice paths with constraints, for various constraints. We show that an approach inspired by optimal transport allows us to bound efficiently the mixing time of the associated Markov chain. The algorithm is robust and easy to implement, and samples an "almost" uniform path of length $n$ in $n^{3+\eps}$ steps. This bound makes use of a certain contraction property of the Markov chain, and is also used to derive a bound for the running time of Propp-Wilson's CFTP algorithm.

http://arxiv.org/abs/1002.1183

9760. Universal Gaussian fluctuations of non-Hermitian matrix ensembles: from weak convergence to almost sure CLTs

Author(s): Ivan Nourdin (PMA) and Giovanni Peccati

Abstract: In the paper [25], written in collaboration with Gesine Reinert, we proved a universality principle for the Gaussian Wiener chaos. In the present work, we aim at providing an original example of application of this principle in the framework of random matrix theory. More specifically, by combining the result in [25] with some combinatorial estimates, we are able to prove multi-dimensional central limit theorems for the spectral moments (of arbitrary degrees) associated with random matrices with real-valued i.i.d. entries, satisfying some appropriate moment conditions. Our approach has the advantage of yielding, without extra effort, bounds over classes of smooth (i.e., thrice differentiable) functions, and it allows to deal directly with discrete distributions. As a further application of our estimates, we provide a new "almost sure central limit theorem", involving logarithmic means of functions of vectors of traces.

http://arxiv.org/abs/1002.1212

9761. A New Phase Transition for Local Delays in MANETs

Author(s): Fran\c{c}ois Baccelli (INRIA Rocquencourt) and Bartek Blaszczyszyn (INRIA Rocquencourt)

Abstract: We consider Mobile Ad-hoc Network (MANET) with transmitters located according to a Poisson point in the Euclidean plane, slotted Aloha Medium Access (MAC) protocol and the so-called outage scenario, where a successful transmission requires a Signal-to-Interference-and-Noise (SINR) larger than some threshold. We analyze the local delays in such a network, namely the number of times slots required for nodes to transmit a packet to their prescribed next-hop receivers. The analysis depends very much on the receiver scenario and on the variability of the fading. In most cases, each node has finite-mean geometric random delay and thus a positive next hop throughput. However, the spatial (or large population) averaging of these individual finite mean-delays leads to infinite values in several practical cases, including the Rayleigh fading and positive thermal noise case. In some cases it exhibits an interesting phase transition phenomenon where the spatial average is finite when certain model parameters are below a threshold and infinite above. We call this phenomenon, contention phase transition. We argue that the spatial average of the mean local delays is infinite primarily because of the outage logic, where one transmits full packets at time slots when the receiver is covered at the required SINR and where one wastes all the other time slots. This results in the "RESTART" mechanism, which in turn explains why we have infinite spatial average. Adaptive coding offers a nice way of breaking the outage/RESTART logic. We show examples where the average delays are finite in the adaptive coding case, whereas they are infinite in the outage case.

http://arxiv.org/abs/1002.0855

9762. Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation

Author(s): Lung-Chi Chen and Akira Sakai

Abstract: We consider random walk and self-avoiding walk whose 1-step distribution is given by D, and oriented percolation whose bond-occupation probability is proportional to D. Suppose that D(x) decays as |x|^{-d-a} with a>0. For random walk in any dimension and for self-avoiding walk and critical/subcritical oriented percolation above the common upper-critical dimension 2min{a,2}, we prove large-t asymptotics of the gyration radius, which is the average end-to-end distance of random walk/self-avoiding walk of length t or the average spatial size of an oriented-percolation cluster at time t. This proves the conjecture for long-range self-avoiding walk by Heydenreich and for long-range oriented percolation in Chen and Sakai (2009).

http://arxiv.org/abs/1002.0875

9763. On the Existence and Uniqueness of Solutions to Stochastic Differential Equations Driven by G-Brownian Motion with Integral-Lipschitz Coefficients

Author(s): Xuepeng Bai and Yiqing Lin

Abstract: In this paper, we study the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion with an integral-Lipschitz condition for the coefficients.

http://arxiv.org/abs/1002.1046

9764. Crowding of Brownian spheres

Author(s): Krzysztof Burdzy and Soumik Pal and Jason Swanson

Abstract: We study two models consisting of reflecting one-dimensional Brownian "particles" of positive radius. We show that the stationary empirical distributions for the particle systems do not converge to the harmonic function for the generator of the individual particle process, unlike in the case when the particles are infinitely small.

http://arxiv.org/abs/1002.1057

9765. Exit times in non-Markovian drifting continuous-time random walk processes

Author(s): Miquel Montero and Javier Villarroel

Abstract: By appealing to renewal theory we determine the equations that the mean exit time of a continuous-time random walk with drift satisfies both when the present coincides with a jump instant or when it does not. Particular attention is paid to the corrections ensuing from the non-Markovian nature of the process. We show that when drift and jumps have the same sign the relevant integral equations can be solved in closed form. The case when holding times have the classical Erlang distribution is considered in detail.

http://arxiv.org/abs/1002.0571

9766. Gibbs Random Graphs

Author(s): Pablo A. Ferrari and Eugene A. Pechersky and Valentin V. Sisko and Anatoly A. Yambartsev

Abstract: Consider a discrete locally finite subset $\Gamma$ of $R^d$ and the complete graph $(\Gamma,E)$, with vertices $\Gamma$ and edges $E$. We consider Gibbs measures on the set of sub-graphs with vertices $\Gamma$ and edges $E'\subset E$. The Gibbs interaction acts between open edges having a vertex in common. We study percolation properties of the Gibbs distribution of the graph ensemble. The main results concern percolation properties of the open edges in two cases: (a) when the $\Gamma$ is a sample from homogeneous Poisson process and (b) for a fixed $\Gamma$ with exponential decay of connectivity.

http://arxiv.org/abs/1002.0610

9767. Series representations and asymptotic expansions for the density of the supremum of a stable process

Author(s): Alexey Kuznetsov

Abstract: We derive explicit asymptotic expansions of the density of the supremum of a strictly stable process when the index $\alpha$ is not rational. In the case when parameters $\alpha$ and $\rho=\p(X_1>0)$ satisfy $\rho+k=l/\alpha$ for some integers $k,l \ge 1$ we prove that these asymptotic expansions are in fact convergent series representations of the density of supremum.

http://arxiv.org/abs/1002.0614

9768. Small Time Chung Type LIL for L\'evy Processes at Zero

Author(s): Frank Aurzada and Leif Doering and and Mladen Savov

Abstract: We prove Chung-type laws of the iterated logarithm for general L\'evy processes at zero. In particular, we provide a tool to translate small deviation estimates directly into laws of the iterated logarithm without any loss of constants nor any extra conditions. This reveals new laws of the iterated logarithm for L\'evy processes at small times in many concrete examples. In some cases, exotic norming functions are derived.

http://arxiv.org/abs/1002.0675

9769. Applications of the graphs to the Generalized Ornstein-Uhlenbeck process

Author(s): Boubaker Smii

Abstract: We consider the generalized Ornstein- Uhlenbeck equation $\partial_t X=-m X_t+\eta$. In this paper We construct the L\'evy noise $\eta$. The generalized Ornstein- Uhlenbeck process $X_t$ will be represented by a special types of graphs called rooted trees with two types of leaves.

http://arxiv.org/abs/1002.0744

9770. Fractional L\'evy processes by compact interval integral transformation

Author(s): Heikki Tikanm\"aki

Abstract: We define fractional L\'evy processes by two different integral transformations by taking integral representation of fractional Brownian motion and replacing the driving Brownian motion by more general square integrable L\'evy process. The definition using infinitely supported transformation kernel is well known in the literature but the definition by compact interval representation is new to the best of my knowledge in this setup. We prove that the processes defined by different transformations do not have the same finite dimensional distributions in general, even though it is the case in fractional Brownian motion setup. Hovever, we prove a connection between the two concepts. We consider different properties of fractional L\'evy processes by compact interval transformation and compare them to the properties of fractional L\'evy processes by infinite interval transformation. We also consider financial applications and represent a no-arbitrage theorem for a model including fractional L\'evy processes by any of the two integral transformations.

http://arxiv.org/abs/1002.0780

9771. Metastability in communication networks

Author(s): D. Tibi

Abstract: Two models of loss networks, introduced by Gibbens et al. and by Antunes et al., are known to exhibit a mean field limiting regime with several stable equilibria. This paper first provides an interpretation of the Lyapunov function given by Antunes et al., in terms of entropy dissipation. The argument extends to another, similar but closed model. The two main models are next reexamined in the light of Freidlin and Wentzell's large deviation approach of randomly perturbed dynamical systems. Assuming that some of their results still hold under slightly relaxed conditions, the metastability property is derived for both systems. The Lyapunov function of the second model is then identified with the quasipotential associated with a slightly modified, asymptotically reversible, Markovian perturbation of the same dynamical system.

http://arxiv.org/abs/1002.0796

9772. Exit times in non-Markovian drifting continuous-time random walk processes

Author(s): Miquel Montero and Javier Villarroel

Abstract: By appealing to renewal theory we determine the equations that the mean exit time of a continuous-time random walk with drift satisfies both when the present coincides with a jump instant or when it does not. Particular attention is paid to the corrections ensuing from the non-Markovian nature of the process. We show that when drift and jumps have the same sign the relevant integral equations can be solved in closed form. The case when holding times have the classical Erlang distribution is considered in detail.

http://www.arxiv.org

9773. Fluctuations for the Ginzburg-Landau $\nabla \phi$ Interface Model on a Bounded Domain

Author(s): Jason Miller

Abstract: The object of our study is the massless field on $D_n = D \cap \tfrac{1}{n} \Z^2$, where $D \subseteq \R^2$ is a bounded domain with smooth boundary, with Hamiltonian $\CH(h) = \sum_{x \sim y} \CV(h(x) - h(y))$ and $h(x) = f(x)$ if $x \in \partial D_n$ for a given continuous function $f \colon \R^2 \to \R$. The interaction $\CV$ is assumed to be symmetric, strictly convex, and have bounded second derivatives. This is a general model for a $(2+1)$-dimensional effective interface where $h$ represents the height. We prove that linear functionals of $h$ converge in the limit to a Gaussian free field on $D$, the standard Gaussian with respect to the Dirichlet inner product $(f,g)_\nabla = \int_D \nabla f \cdot \nabla g$. The main step in the proof is to establish a general estimate that serves to quantify the degree to which the presence of boundary conditions affect the behavior of the model. In particular, our estimate implies that $\E h(x)$ converges to the harmonic extension of $f$ from $\partial D$ to $D$. In a subsequent article, we will employ the tools developed here to resolve a conjecture made by Sheffield that the zero contour lines of $h$ are asymptotically described by a family of conformally invariant random curves which are variants of SLE(4).

http://arxiv.org/abs/1002.0381

9774. The maximum of Brownian motion with parabolic drift

Author(s): Svante Janson and Guy Louchard and Anders Martin-L\"of

Abstract: We study the maximum of a Brownian motion with a parabolic drift; this is a random variable that often occurs as a limit of the maximum of discrete processes whose expectations have a maximum at an interior point. We give series expansions and integral formulas for the distribution and the first two moments, together with numerical values to high precision.

http://arxiv.org/abs/1002.0497

9775. Convergence of an Adaptive Approximation Scheme for the Wiener Process

Author(s): Mats Brod\'en and Magnus Wiktorsson

Abstract: The problem of approximating/tracking the value of a Wiener process is considered. The discretization points are placed at times when the value of the process differs from the approximation by some amount, here denoted by eta. It is found that the limiting difference, as eta goes to 0, between the approximation and the value of the process normalized with eta converges in distribution to a triangularly distributed random variable.

http://arxiv.org/abs/1002.0528

9776. On some Bayesian nonparametric estimators for species richness under two-parameter Poisson-Dirichlet priors

Author(s): Annalisa Cerquetti

Abstract: We present an alternative approach to the Bayesian nonparametric analysis of conditional species richness under two-parameter Poisson Dirichlet priors. We rely on a known characterization by deletion of classes property and on results for Beta-Binomial distributions. Besides leading to simplified and much more direct proofs, our proposal provides a new scale mixture representation of the conditional asymptotic law.

http://arxiv.org/abs/1002.0535

9777. On Stochastic generalized functions

Author(s): Pedro Catuogno and Christian Olivera

Abstract: We introduced a new algebra of stochastic generalized functions which contains to the space of stochastic distributions G, [25]. As an application, we prove existence and uniqueness of the solution of a stochastic Cauchy problem involving singularities.

http://arxiv.org/abs/1002.0454

9778. Wright-Fisher model with negative mutation rates

Author(s): Soumik Pal

Abstract: We study a family of multidimensional diffusions taking values in the unit simplex of vectors with non-negative coordinates that add up to one. The family of processes satisfy stochastic differential equations which are similar to the ones for the classical Wright-Fisher model, except that the "mutation rates" are now nonpositive. This model, suggested by Aldous, appears in the study of a conjectured diffusion limit for a Markov chain on Cladograms. The striking feature of these models is that the boundary is not reflecting, and we kill the process once it hits the boundary. We derive the explicit exit distribution from the simplex, and probabilistic bounds on the exit time. We also prove that these processes can be viewed as a "stochastic time-reversal" of a Wright-Fisher process of increasing dimensions and conditioned at a random time. A key idea in our proofs is a skew-product construction using certain one-dimensional diffusions called Bessel-square processes of negative dimensions which have been recently introduced by Going-Jaeschke and Yor.

http://arxiv.org/abs/1002.0159

9779. BS$\Delta$Es and BSDEs with non-Lipschitz drivers: comparison, convergence and robustness

Author(s): Patrick Cheridito and Mitja Stadje

Abstract: We provide existence results and comparison principles for solutions of backward stochastic difference equations (BS$\Delta$Es) and then prove convergence of these to solutions of backward stochastic differential equations (BSDEs) when the mesh size of the time-discretizaton goes to zero. The BS$\Delta$Es and BSDEs are governed by drivers $f^N(t,\omega,y,z)$ and $f(t,\omega,y,z),$ respectively. The new feature of this paper is that they may be non-Lipschitz in $z$. For the convergence results it is assumed that the BS$\Delta$Es are based on $d$-dimensional random walks $W^N$ approximating the $d$-dimensional Brownian motion $W$ underlying the BSDE and that $f^N$ converges to $f$. Conditions are given under which for any terminal condition $\xi$, there exist terminal conditions $\xi^N$ for the sequence of BS$\Delta$Es, converging to $\xi$ in $L^2$, such that for the solutions $Y^N$ and $Y$ of the corresponding BS$\Delta$Es and the limiting BSDE one has $\sup_{0\le t\le T} |Y^N_t - Y_t| \to 0$ in $L^2$. An important special case is when $f^N(t,\omega,y,z)$ and $f(t,\omega,y,z)$ are convex in $z.$ We show that in this situation, $\sup_{0\le t\le T} |Y^N_t - Y_t| \to 0$ in $L^2$ for every sequence of discrete terminal conditions $\xi^N$ converging to $\xi$ in $L^2$. As a consequence, one obtains that the BSDE is robust in the sense that if $(W^N,\xi^N)$ is close to $(W,\xi)$ in distribution, then $Y^N$ is close to $Y$ in distribution too.

http://arxiv.org/abs/1002.0175

9780. Convergence of U-statistics for interacting particle systems

Author(s): P. Del Moral and F. Patras and S. Rubenthaler

Abstract: The convergence of U-statistics has been intensively studied for estimators based on families of i.i.d. random variables and variants of them. In most cases, the independence assumption is crucial [Lee90, de99]. When dealing with Feynman-Kac and other interacting particle systems of Monte Carlo type, one faces a new type of problem. Namely, in a sample of N particles obtained through the corresponding algorithms, the distributions of the particles are correlated -although any finite number of them is asymptotically independent with respect to the total number N of particles. In the present article, exploiting the fine asymptotics of particle systems, we prove convergence theorems for U-statistics in this framework.

http://arxiv.org/abs/1002.0224

9781. Ergodicity for infinite particle systems with locally conserved quantities

Author(s): J. Inglis and M. Neklyudov and B. Zegarlinski

Abstract: We analyse certain degenerate infinite dimensional sub-elliptic generators and obtain estimates on the long-time behaviour of the corresponding Markov semigroups.

http://arxiv.org/abs/1002.0282

9782. Non-Equilibrium Statistical Physics of Currents in Queuing Networks

Author(s): Vladimir Y.Chernyak and Michael Chertkov and David A. Goldberg and Konstantin Turitsyn

Abstract: A stable open queuing network is considered as a steady non-equilibrium system of interacting particles. The network is completely specified by its underlying graphical structure, type of interaction at each node, and the Poisson transition rates between nodes. For such systems we identify two regimes in which the system may operate depending on the value of currents accumulated on the graph edges over time, large compared to the system correlation time scale. In the first regime of moderate currents, the large-deviation distribution of currents is universal (independent of the interaction details), and the system behaves in an "uncongested" mode. In the second regime of larger currents, the large-deviation current distribution is sensitive to interaction details, and the system is in a "congested" mode. The transition between the two regimes can be described as a dynamical second order phase transition. We illustrate these ideas using a simple, yet non-trivial, example of a single node with feedback.

http://arxiv.org/abs/1001.5454

9783. Scalar curvature and $Q$-curvature of random metrics

Author(s): Yaiza Canzani and Dmitry Jakobson and Igor Wigman

Abstract: We study Gauss curvature for random Riemannian metrics on a compact surface, lying in a fixed conformal class; our questions are motivated by comparison geometry. Next, analogous questions are considered for the scalar curvature in dimension $n>2$, and for the $Q$-curvature of random Riemannian metrics.

http://arxiv.org/abs/1002.0030

9784. Left and right convergence of graphs with bounded degree

Author(s): Christian Borgs and Jennifer Chayes and Jeff Kahn and L\'aszl\'o Lov\'asz

Abstract: The theory of convergent graph sequences has been worked out in two extreme cases, dense graphs and bounded degree graphs. One can define convergence in terms of counting homomorphisms from fixed graphs into members of the sequence (left-convergence), or counting homomorphisms into fixed graphs (right-convergence). Under appropriate conditions, these two ways of defining convergence was proved to be equivalent in the dense case by Borgs, Chayes, Lov\'asz, S\'os and Vesztergombi. In this paper a similar equivalence is established in the bounded degree case. In terms of statistical physics, the implication that left convergence implies right convergence means that for a left-convergent sequence, partition functions of a large class of statistical physics models converge. The proof relies on techniques from statistical physics, like cluster expansion and Dobrushin Uniqueness.

http://arxiv.org/abs/1002.0115

9785. Kalman-Bucy filter and SPDEs with growing lower-order coefficients in $W^{1}_{p}$ spaces without weights

Author(s): N.V. Krylov

Abstract: We consider divergence form uniformly parabolic SPDEs with VMO bounded leading coefficients, bounded coefficients in the stochastic part, and possibly growing lower-order coefficients in the deterministic part. We look for solutions which are summable to the $p$th power, $p\geq2$, with respect to the usual Lebesgue measure along with their first-order derivatives with respect to the spatial variable. Our methods allow us to include Zakai's equation for the Kalman-Bucy filter into the general filtering theory.

http://arxiv.org/abs/1002.0306

9786. Packing and Hausdorff measures of stable trees

Author(s): Thomas Duquesne (PMA)

Abstract: In this paper we discuss Hausdorff and packing measures of random continuous trees called stable trees. Stable trees form a specific class of L\'evy trees (introduced by Le Gall and Le Jan in 1998) that contains Aldous's continuum random tree (1991) which corresponds to the Brownian case. We provide results for the whole stable trees and for their level sets that are the sets of points situated at a given distance from the root. We first show that there is no exact packing measure for levels sets. We also prove that non-Brownian stable trees and their level sets have no exact Hausdorff measure with regularly varying gauge function, which continues previous results from a joint work with J-F Le Gall (2006).

http://arxiv.org/abs/1001.5329

9787. Universality of slow decorrelation in KPZ growth

Author(s): I. Corwin and P.L. Ferrari and S. Peche

Abstract: We demonstrate that, under minimal hypothesis, a wide class of growth models diplays a phenomenon known as slow decorrelation, where along certain characteristic directions the range of correlation for fluctuations of the growth surface height is much longer than other directions. We apply this result to certain models known to be in the Kardar-Parisi-Zhang (KPZ) universality class in 1+1 dimension for which the necessary hypothesis holds. These models are the totally asymmetric simple exclusion process (TASEP), last passage percolation (LPP), and the polynuclear growth (PNG) model. Utilizing the slow decorrelation of fluctuations in these models we are able to extend known fluctuation limit process results away from the fixed curves on which they were proved, to general space-time curves. Using the monotonicity of the basic coupling we additionally prove that the partially asymmetric simple exclusion process (PASEP) displays slow decorrelation.

http://arxiv.org/abs/1001.5345

9788. The shortest distance in random multi-type intersection graphs

Author(s): A. D. Barbour and G. Reinert

Abstract: Using an associated branching process as the basis of our approximation, we show that typical inter-point distances in a multitype random intersection graph have a defective distribution, which is well described by a mixture of translated and scaled Gumbel distributions, the missing mass corresponding to the event that the vertices are not in the same component of the graph.

http://arxiv.org/abs/1001.5357

9789. Long time behaviour in a model of microtubule growth

Author(s): O.Hryniv and M.Menshikov

Abstract: We study a continuous time stochastic process on strings made of two types of particles, whose dynamics mimics the behaviour of microtubules in a living cell; namely, the strings evolve via a competition between (local) growth/shrinking as well as (global) hydrolysis processes. We give a complete characterization of the phase diagram of the model, and derive several criteria of the transient and recurrent regimes for the underlying stochastic process.

http://arxiv.org/abs/1001.5469

9790. On uniqueness of mild solutions for dissipative stochastic evolution equations

Author(s): Carlo Marinelli and Michael R\"ockner

Abstract: In the semigroup approach to stochastic evolution equations, the fundamental issue of uniqueness of mild solutions is often "reduced" to the much easier problem of proving uniqueness for strong solutions. This reduction is usually carried out in a formal way, without really justifying why and how one can do that. We provide sufficient conditions for uniqueness of mild solutions to a broad class of semilinear stochastic evolution equations with coefficients satisfying a monotonicity assumption.

http://arxiv.org/abs/1001.5413

9791. Scalar conservation laws with stochastic forcing

Author(s): Arnaud Debussche (IRMAR) and Julien Vovelle (ICJ)

Abstract: We show that the Cauchy Problem for a randomly forced, periodic multi-dimensional scalar first-order conservation law with additive or multiplicative noise is well-posed: it admits a unique solution, characterized by a kinetic formulation of the problem, which is the limit of the solution of the stochastic parabolic approximation.

http://arxiv.org/abs/1001.5415

9792. Hidden Regular Variation: Detection and Estimation

Author(s): Abhimanyu Mitra and Sidney I. Resnick

Abstract: Hidden regular variation defines a subfamily of distributions satisfying multivariate regular variation on $\mathbb{E} = [0, \infty]^d \backslash \{(0,0, ..., 0) \} $ and models another regular variation on the sub-cone $\mathbb{E}^{(2)} = \mathbb{E} \backslash \cup_{i=1}^d \mathbb{L}_i$, where $\mathbb{L}_i$ is the $i$-th axis. We extend the concept of hidden regular variation to sub-cones of $\mathbb{E}^{(2)}$ as well. We suggest a procedure of detecting the presence of hidden regular variation, and if it exists, propose a method of estimating the limit measure exploiting its semi-parametric structure. We exhibit examples where hidden regular variation yields better estimates of probabilities of risk sets.

http://arxiv.org/abs/1001.5058

9793. Fluctuations of the occupation times for branching system starting from infinitely divisible point processes

Author(s): Piotr Milos

Abstract: In the paper the rescaled occupation time fluctuation process of a certain empirical system is investigated. The system consists of particles evolving independently according to \alpha-stable motion in R^d, \alpha0. We study how the limit behaviour of the fluctuations of the occupation time depends on the \emph{initial particle configuration}. We obtain a functional central limit theorem for a vast class of infinitely divisible distributions. Our findings extend and put in a unified setting results which previously seemed to be disconnected. The limit processes form a one dimensional family of long-range dependance centred Gaussian processes.

http://arxiv.org/abs/1001.5142

9794. A representation formula for the Freidlin-Wentzell functional on the one dimensional torus

Author(s): A. Faggionato and D. Gabrielli

Abstract: Inspired by some recent results on fluctuation theory for piecewise deterministic Markov processes, we consider a generic diffusion on the 1D torus and give a simple representation formula for the large deviation rate functional of its invariant probability measure, in the limit of vanishing noise. Previously, this rate functional had been characterized by M.I. Freidlin and A.D. Wentzell as solution of a rather complex optimization problem. We discuss this last problem in full generality and show that it leads to our formula. Finally, we discuss some geometric and regularity properties of the rate functional. In particular, we prove a universality result showing that the rate functional is a viscosity solution of the stationary Hamilton--Jacobi equation associated to any Hamiltonian H satisfying weak suitable conditions.

http://arxiv.org/abs/1001.5160

9795. A Laplace principle for a stochastic wave equation in spatial dimension three

Author(s): V\'ictor Ortiz-L\'opez and Marta Sanz-Sol\'e

Abstract: We consider a stochastic wave equation in spatial dimension three, driven by a Gaussian noise, white in time and with a stationary spatial covariance. The free terms are nonlinear with Lipschitz continuous coefficients. Under suitable conditions on the covariance measure, Dalang and Sanz-Sol\'e [Memoirs of the AMS, Vol 199, 2009] have proved the existence of a random field solution with H\"older continuous sample paths, jointly in both arguments, time and space. By perturbing the driving noise with a multiplicative parameter $\varepsilon\in]0,1]$, a family of probability laws corresponding to the respective solutions to the equation is obtained. Using the weak convergence approach to large deviations developed in [P. Dupuis, R. S. Ellis, 1997], we prove that this family satisfies a Laplace principle in the H\"older norm.

http://arxiv.org/abs/1001.5228

9796. Large deviations for slow-fast stochastic partial differential equations

Author(s): Wei Wang and A. J. Roberts and Jinqiao Duan

Abstract: A large deviation principle is derived for stochastic partial differential equations with slow-fast components. The result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small Gaussian perturbation. This also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow-fast stochastic partial differential equations.

http://arxiv.org/abs/1001.4826

9797. Stochastic Approximation, Cooperative Dynamics and Supermodular Games

Author(s): Michel Bena\"im (UNINE) and Mathieu Faure (UNINE)

Abstract: This paper considers a stochastic approximation algorithm, with decreasing step size and martingale difference noise. Under very mild assumptions, we prove the non convergence of this process toward a certain class of repulsive sets for the associated ordinary differential equation (ODE). We then use this result to derive the convergence of the process when the ODE is cooperative in the sense of [Hirsch, 1985]. In particular, this allows us to extend significantly the main result of [Hofbauer and Sandholm, 2002] on the convergence of stochastic fictitious play in supermodular games.

http://arxiv.org/abs/1001.4871

9798. The asymptotic behavior of densities related to the supremum of a stable process

Author(s): R. A. Doney and M. S. Savov

Abstract: If $X$ is a stable process of index $\alpha\in(0,2)$ whose L\'{e}vy measure has density $cx^{-\alpha-1}$ on $(0,\infty)$, and $S_1=\sup_{0x)\backsim A\alpha ^{-1}x^{-\alpha}$ as $x\to\infty$ and $P(S_1\leq x)\backsim B\alpha^{-1}\rho^{-1}x^{\alpha\rho}$ as $x\downarrow0$. [Here $\rho =P(X_1>0)$ and $A$ and $B$ are known constants.] It is also known that $S_1$ has a continuous density, $m$ say. The main point of this note is to show that $m(x)\backsim Ax^{-(\alpha+1)}$ as $x\to\infty$ and $m(x)\backsim Bx^{\alpha\rho-1}$ as $x\downarrow0$. Similar results are obtained for related densities.

http://arxiv.org/abs/1001.4872

9799. Properties of hitting times for $G$-martingale

Author(s): Yongsheng Song

Abstract: In this article, we consider the properties of hitting times for $G$-martingale and the stopped processes. We prove that the stopped processes for $G$-martingales are still $G$-martingales and that the hitting times for a class of $G$-martingales including $G$-Brownian motion are quasi-continuous.

http://arxiv.org/abs/1001.4907

9800. Limit theorems for the number of occupied boxes in the Bernoulli sieve

Author(s): Alexander Gnedin and Alexander Iksanov and and Alexander Marynych

Abstract: The Bernoulli sieve is a version of the classical `balls-in-boxes' occupancy scheme, in which random frequencies of infinitely many boxes are produced by a multiplicative renewal process, also known as the residual allocation model or stick-breaking. We focus on the number $K_n$ of boxes occupied by at least one of $n$ balls, as $n\to\infty$. A variety of limiting distributions for $K_n$ is derived from the properties of associated perturbed random walks. Refining the approach based on the standard renewal theory we remove a moment constraint to cover the cases left open in previous studies.

http://arxiv.org/abs/1001.4920

9801. Limit law for some modified ergodic sums

Author(s): Jean-Pierre Conze (IRMAR) and St\'ephane Le Borgne (IRMAR)

Abstract: An example due to Erdos and Fortet shows that, for a lacunary sequence of integers (q_n) and a trigonometric polynomial f, the asymptotic distribution of normalized sums of f(q_k x) can be a mixture of gaussian laws. Here we give a generalization of their example interpreted as the limiting behavior of some modified ergodic sums in the framework of dynamical systems.

http://arxiv.org/abs/1001.4862

9802. The McShane integral in weakly compactly generated spaces

Author(s): Antonio Avil\'es and Grzegorz Plebanek and Jos\'e Rodr\'iguez

Abstract: Di Piazza and Preiss asked whether every Pettis integrable function defined on [0,1] and taking values in a weakly compactly generated Banach space is McShane integrable. In this paper we answer this question in the negative.

http://arxiv.org/abs/1001.4896

9803. Percolation on self-dual polygon configurations

Author(s): Bela Bollobas and Oliver Riordan

Abstract: Recently, Scullard and Ziff noticed that a broad class of planar percolation models are self-dual under a simple condition that, in a parametrized version of such a model, reduces to a single equation. They state that the solution of the resulting equation gives the critical point. However, just as in the classical case of bond percolation on the square lattice, self-duality is simply the starting point: the mathematical difficulty is precisely showing that self-duality implies criticality. Here we do so for a generalization of the models considered by Scullard and Ziff. In these models, the states of the bonds need not be independent; furthermore, increasing events need not be positively correlated, so new techniques are needed in the analysis. The main new ingredients are a generalization of Harris's Lemma to products of partially ordered sets, and a new proof of a type of Russo-Seymour-Welsh Lemma with minimal symmetry assumptions.

http://arxiv.org/abs/1001.4674

9804. On Beta-Product Convolutions

Author(s): Enkelejd Hashorva

Abstract: Let R be a positive random variable independent of S which is beta distributed. In this paper we are interested on the relation between the distribution function of $R$ and that of RS. For this model we derive first some distributional properties, and then investigate the lower tail asymptotics of RS when R is regularly varying at 0, and vice-versa. The applications we present in this paper concern a) the simplicity of Dirichlet distributions, b) asymptotics of the sample minima of elliptical distributions, and c) the effect of the scaling on the asymptotics of aggregated risks.

http://arxiv.org/abs/1001.4684

9805. Invariance principle for the random conductance model with unbounded conductances

Author(s): M. T. Barlow and J.-D. Deuschel

Abstract: We study a continuous time random walk $X$ in an environment of i.i.d. random conductances $\mu_e\in[1,\infty)$. We obtain heat kernel bounds and prove a quenched invariance principle for $X$. This holds even when ${\mathbb{E}}\mu_e=\infty$.

http://arxiv.org/abs/1001.4702

9806. A shape theorem and semi-infinite geodesics for the Hammersley model with random weights

Author(s): E.A. Cator and L.P.R. Pimentel

Abstract: In this paper we will prove a shape theorem for the last passage percolation model on a two dimensional $F$-compound Poisson process, called the Hammersley model with random weights. We will also provide diffusive upper bounds for shape fluctuations. Finally we will indicate how these results can be used to prove existence and coalescence of semi-infinite geodesics in some fixed direction $\alpha$, following an approach developed by Newman and co-authors, and applied to the classical Hammersley process by W\"uthrich. These results will be crucial in the development of an upcoming paper on the relation between Busemann functions and equilibrium measures in last passage percolation models.

http://arxiv.org/abs/1001.4706

9807. On the existence and position of the farthest peaks of a family of stochastic heat and wave equations

Author(s): Daniel Conus and Davar Khoshnevisan

Abstract: We study a family of non-linear stochastic heat equations in (1+1) dimensions, driven by the generator of a L\'evy process and space-time white noise. We assume that the underlying L\'evy process has finite exponential moments in a neighborhood of the origin and that the initial condition has exponential decay at infinity. Then we prove that under natural conditions on the non-linearity: (i) The absolute moments of the solution to our stochastic heat equation grow exponentially with time; and (ii) The distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. Very little else seems to be known about the location of the high peaks of the solution to the non-linear stochastic heat equation. Finally, we show that these results extend to the stochastic wave equation driven by Laplacian.

http://arxiv.org/abs/1001.4759

9808. Formulas for ASEP with Two-Sided Bernoulli Initial Condition

Author(s): Craig A. Tracy and Harold Widom

Abstract: For the asymmetric simple exclusion process on the integer lattice with two-sided Bernoulli initial condition, we derive exact formulas for the following quantities: (1) the probability that site x is occupied at time t; (2) a correlation function, the probability that site 0 is occupied at time 0 and site x is occupied at time t; (3) the distribution function for the total flux across 0 at time t and its exponential generating function.

http://arxiv.org/abs/1001.4766

9809. Joint distribution of the process and its sojourn time for pseudo-processes governed by high-order heat equation

Author(s): Valentina Cammarota and Aime 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 the sojourn time $T(t)$ in the interval $[0,+\infty)$ up to a fixed time $t$ for this pseudo-process. We provide explicit expressions for the joint distribution of the couple $(T(t),X(t))$.

http://arxiv.org/abs/1001.4201

9810. Optimal tuning of the Hybrid Monte-Carlo Algorithm

Author(s): Alexandros Beskos and Natesh S. Pillai and Gareth O. Roberts and Jesus M. Sanz-Serna, Andrew M. Stuart

Abstract: We investigate the properties of the Hybrid Monte-Carlo algorithm (HMC) in high dimensions. HMC develops a Markov chain reversible w.r.t. a given target distribution $\Pi$ by using separable Hamiltonian dynamics with potential $-\log\Pi$. The additional momentum variables are chosen at random from the Boltzmann distribution and the continuous-time Hamiltonian dynamics are then discretised using the leapfrog scheme. The induced bias is removed via a Metropolis-Hastings accept/reject rule. In the simplified scenario of independent, identically distributed components, we prove that, to obtain an $\mathcal{O}(1)$ acceptance probability as the dimension $d$ of the state space tends to $\infty$, the leapfrog step-size $h$ should be scaled as $h= l \times d^{-1/4}$. Therefore, in high dimensions, HMC requires $\mathcal{O}(d^{1/4})$ steps to traverse the state space. We also identify analytically the asymptotically optimal acceptance probability, which turns out to be 0.651 (to three decimal places). This is the choice which optimally balances the cost of generating a proposal, which {\em decreases} as $l$ increases, against the cost related to the average number of proposals required to obtain acceptance, which {\em increases} as $l$ increases.

http://arxiv.org/abs/1001.4460

9811. Emergence of Randomness and Arrow of Time in Quantum Walks

Author(s): Yutaka Shikano and Kota Chisaki and Etsuo Segawa and Norio Konno

Abstract: Quantum walks are powerful tools not only to construct the quantum speedup algorithms but also to describe specific models in physical processes. Furthermore, the discrete time quantum walk has been experimentally realized in various setups. We apply the concept of the quantum walk to the problems in quantum foundations. We show that randomness and the arrow of time in the quantum walk gradually emerge by periodic projective measurements from the mathematically obtained limit distribution under the time scale transformation.

http://arxiv.org/abs/1001.3989

9812. On the Zero-Type property and mixing of Bernoulli shifts

Author(s): Zemer Kosloff

Abstract: We extend the notion of zero-type to the whole class of non-singular transformations and then prove that every non-singular Bernoulli shift is either zero-type or there is an equivalent invariant probability.

http://arxiv.org/abs/1001.4261

9813. On the Vershik-Kerov Conjecture Concerning the Shannon-Macmillan-Breiman Theorem for the Plancherel Family of Measures on the Space of Young Diagrams

Author(s): Alexander I. Bufetov

Abstract: Vershik and Kerov conjectured in 1985 that suitably normalized dimensions of irreducible representations of finite symmetric groups converge to a constant with respect to the Plancherel family of measures on the space of Young diagrams. The main result of this paper is the proof of the Vershik-Kerov conjecture. The argument is based on the methods of Borodin, Okounkov and Olshanski.

http://arxiv.org/abs/1001.4275

9814. Martingale representation for Poisson processes with applications to minimal variance hedging

Author(s): Guenter Last and Mathew D. Penrose

Abstract: We consider a Poisson process $\eta$ on a measurable space $(\BY,\mathcal{Y})$ equipped with a partial ordering, assumed to be strict almost everwhwere with respect to the intensity measure $\lambda$ of $\eta$. We give a Clark-Ocone type formula providing an explicit representation of square integrable martingales (defined with respect to the natural filtration associated with $\eta$), which was previously known only in the special case, when $\lambda$ is the product of Lebesgue measure on $\R_+$ and a $\sigma$-finite measure on another space $\BX$. Our proof is new and based on only a few basic properties of Poisson processes and stochastic integrals. We also consider the more general case of an independent random measure in the sense of It\^o of pure jump type and show that the Clark-Ocone type representation leads to an explicit version of the Kunita-Watanabe decomposition of square integrable martingales. We also find the explicit minimal variance hedge in a quite general financial market driven by an independent random measure.

http://arxiv.org/abs/1001.3972

9815. A d dimensional nucleation and growth model

Author(s): Raphael Cerf and Francesco Manzo

Abstract: We analyze the relaxation time of a ferromagnetic d dimensional growth model on the lattice. The model is characterized by d param- eters which represent the activation energies of a site, depending on the number of occupied nearest neighbours. This model is a natural generalisation of the model studied by Dehghanpour and Schonmann [DS97a], where the activation energy of a site with more than two occupied neighbours is zero.

http://arxiv.org/abs/1001.3990

9816. Stochastic evolution equations driven by Liouville fractional Brownian motion

Author(s): Zdzislaw Brzezniak and Jan van Neerven and Donna Salopek

Abstract: Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of L(H,E)-valued functions with respect to H-cylindrical Liouville fractional Brownian motions (fBm) with arbitrary Hurst parameter in the interval (0,1). For Hurst parameters in (0,1/2) we show that a function F:(0,T)\to L(H,E) is stochastically integrable with respect to an H-cylindrical Liouville fBm if and only if it is stochastically integrable with respect to an H-cylindrical fBm with the same Hurst parameter. As an application we show that second-order parabolic SPDEs on bounded domains in \mathbb{R}^d, driven by space-time noise which is white in space and Liouville fractional in time with Hurst parameter in (d/4,1) admit mild solution which are H\"older continuous both and space.

http://arxiv.org/abs/1001.4013

9817. Spanning forests and the vector bundle laplacian

Author(s): Richard Kenyon

Abstract: The classical matrix-tree theorem relates the determinant of the combinatorial laplacian on a graph to the number of spanning trees. We generalize this result to laplacians on one- and two-dimensional vector bundles, giving a combinatorial interpretation of their determinants in terms of so-called cycle rooted spanning forests. We construct natural measures on CRSFs for which the edges form a determinantal process. This theory gives a natural generalization of the spanning tree process adapted to graphs embedded on surfaces. We give a number of other applications, for example we compute the probability that a loop-erased random walk on a planar graph between two vertices on the outer boundary passes left of two given faces. This probability can not be computed using the standard Laplacian alone.

http://arxiv.org/abs/1001.4028

9818. Is the minimum value of an option on variance generated by local volatility?

Author(s): Mathias Beiglboeck and Peter Friz and Stefan Sturm

Abstract: We discuss the possibility of obtaining model-free bounds on volatility derivatives, given present market data in the form of a calibrated local volatility model. A counter-example to a wide-spread conjecture is given.

http://arxiv.org/abs/1001.4031

9819. Thermophoresis as persistent random walk

Author(s): A.V. Plyukhin

Abstract: In a simple model of a continuous random walk a particle moves in one dimension with the velocity fluctuating between V and -V. If V is associated with the thermal velocity of a Brownian particle and allowed to be position dependent, the model accounts readily for the particle's drift along the temperature gradient and recovers basic results of the conventional thermophoresis theory.

http://arxiv.org/abs/0903.3584

9820. Stochastic process leading to wave equations in dimensions higher than one

Author(s): A.V. Plyukhin

Abstract: Stochastic processes are proposed whose master equations coincide with classical wave, telegraph, and Klein-Gordon equations. Similar to predecessors based on the Goldstein-Kac telegraph process, the model describes the motion of particles with constant speed and transitions between discreet allowed velocity directions. A new ingredient is that transitions into a given velocity state depend on spatial derivatives of other states populations, rather than on populations themselves. This feature requires the sacrifice of the single-particle character of the model, but allows to imitate the Huygens' principle and to recover wave equations in arbitrary dimensions.

http://arxiv.org/abs/1001.3821

9821. An aging phenomenon for a fragmentation-coagulation process

Author(s): Jean Bertoin (PMA and Dma)

Abstract: We point out that aging occurs for the following simple model of fragmentation-coagulation inspired by Pitman's coalescent random forests. For every $n\in \N$, we consider a uniform random tree with $n$ vertices, and at each step, depending on the outcome of an independent fair coin tossing, either we remove one edge chosen uniformly at random amongst the remaining edges, or we replace one edge chosen uniformly at random amongst the edges which have been removed previously. The process that records the sizes of the tree-components evolves by fragmentation and coagulation. It exhibits aging in the sense that when it is observed after $k$ steps in the regime $k\sim tn+s\sqrt n$ with $t>0$ fixed, it seems to reach a statistical equilibrium as $n\to\infty$; but different values of $t$ yield distinct pseudo-stationary distributions. The approach owes much to the construction by Aldous and Pitman of the standard additive coalescent via Poissonian cuts on the skeleton of a Continuum Random Tree.

http://arxiv.org/abs/1001.3721

9822. Jump-diffusion modeling in emission markets

Author(s): K. Borovkov and G. Decrouez and J. Hinz

Abstract: Mandatory emission trading schemes are being established around the world. Participants of such market schemes are always exposed to risks. This leads to the creation of an accompanying market for emission-linked derivatives. To evaluate the fair prices of such financial products, one needs appropriate models for the evolution of the underlying assets, emission allowance certificates. In this paper, we discuss continuous time diffusion and jump-diffusion models, the latter enabling one to model information shocks that cause jumps in allowance prices. We show that the resulting martingale dynamics can be described in terms of non-linear partial differential and integro-differential equations and use a finite difference method to investigate numerical properties of their discretizations. The results are illustrated by a small numerical study.

http://arxiv.org/abs/1001.3728

9823. Martingale Representation Theorem for the G-expectation

Author(s): H.M. Soner; N. Touzi; J. Zhang

Abstract: This paper considers the nonlinear theory of G-martingales as introduced by Peng. A martingale representation theorem for this theory is proved by using the techniques and the results established in an accompanying paper for the second order stochastic target problems and the second order backward stochastic differential equations. In particular, this representation provides a hedging strategy in a market with an uncertain volatility.

http://arxiv.org/abs/1001.3802

9824. Limit theorem for randomly indexed sequence of random processes

Author(s): Elena Permyakova

Abstract: In this paper is proved the limit theorem for randomly indexed sequence of random processes in the case where sequences of random index and random processes are independent, also the estimation of convergence rate is obtained.

http://arxiv.org/abs/1001.3844

9825. Stochastic firing rate models

Author(s): Jonathan Touboul and Bard Ermentrout and Olivier Faugeras and Bruno Cessac

Abstract: We review a recent approach to the mean-field limits in neural networks that takes into account the stochastic nature of input current and the uncertainty in synaptic coupling. This approach was proved to be a rigorous limit of the network equations in a general setting, and we express here the results in a more customary and simpler framework. We propose a heuristic argument to derive these equations providing a more intuitive understanding of their origin. These equations are characterized by a strong coupling between the different moments of the solutions. We analyse the equations, present an algorithm to simulate the solutions of these mean-field equations, and investigate numerically the equations. In particular, we build a bridge between these equations and Sompolinsky and collaborators approach (1988, 1990), and show how the coupling between the mean and the covariance function deviates from customary approaches.

http://arxiv.org/abs/1001.3872

9826. Functional Regression for General Exponential Families

Author(s): Wei Dou and David Pollard and and Harrison H. Zhou

Abstract: The paper derives a minimax lower bound for rates of convergence for an infinite-dimensional parameter in an exponential family model. An estimator that achieves the optimal rate is constructed by maximum likelihood on finite-dimensional approximations with parameter dimension that grows with sample size.

http://arxiv.org/abs/1001.3742

9827. Secure Communication in Stochastic Wireless Networks

Author(s): Pedro C. Pinto and Joao Barros and Moe Z. Win

Abstract: Information-theoretic security -- widely accepted as the strictest notion of security -- relies on channel coding techniques that exploit the inherent randomness of the propagation channels to significantly strengthen the security of digital communications systems. Motivated by recent developments in the field, this paper aims at a characterization of the fundamental secrecy limits of wireless networks. Based on a general model in which legitimate nodes and potential eavesdroppers are randomly scattered in space, the intrinsically secure communications graph (iS-graph) is defined from the point of view of information-theoretic security. Conclusive results are provided for the local connectivity of the Poisson iS-graph, in terms of node degrees and isolation probabilities. It is shown how the secure connectivity of the network varies with the wireless propagation effects, the secrecy rate threshold of each link, and the noise powers of legitimate nodes and eavesdroppers. Sectorized transmission and eavesdropper neutralization are explored as viable strategies for improving the secure connectivity. Lastly, the maximum secrecy rate between a node and each of its neighbours is characterized, and the case of colluding eavesdroppers is studied. The results help clarify how the spatial density of eavesdroppers can compromise the intrinsic security of wireless networks.

http://arxiv.org/abs/1001.3697

9828. Stochastic Switching Games and Duopolistic Competition in Emissions Markets

Author(s): Michael Ludkovski

Abstract: We study optimal behavior of energy producers under a CO_2 emission abatement program. We focus on a two-player discrete-time model where each producer is sequentially optimizing her emission and production schedules. The game-theoretic aspect is captured through a reduced-form price-impact model for the CO_2 allowance price. Such duopolistic competition results in a new type of a non-zero-sum stochastic switching game on finite horizon. Existence of game Nash equilibria is established through generalization to randomized switching strategies. No uniqueness is possible and we therefore consider a variety of correlated equilibrium mechanisms. We prove existence of correlated equilibrium points in switching games and give a recursive description of equilibrium game values. A simulation-based algorithm to solve for the game values is constructed and a numerical example is presented.

http://arxiv.org/abs/1001.3455

9829. On the inference of large phylogenies with long branches: How long is too long?

Author(s): Elchanan Mossel and Sebastien Roch and Allan Sly

Abstract: Recent work has highlighted deep connections between sequence-length requirements for high-probability phylogeny reconstruction and the related problem of the estimation of ancestral sequences. In [Daskalakis et al.'09], building on the work of [Mossel'04], a tight sequence-length requirement was obtained for the CFN model. In particular the required sequence length for high-probability reconstruction was shown to undergo a sharp transition (from $O(\log n)$ to $\hbox{poly}(n)$, where $n$ is the number of leaves) at the "critical" branch length $\critmlq$ (if it exists) of the ancestral reconstruction problem. Here we consider the GTR model. For this model, recent results of [Roch'09] show that the tree can be accurately reconstructed with sequences of length $O(\log(n))$ when the branch lengths are below $\critksq$, known as the Kesten-Stigum (KS) bound. Although for the CFN model $\critmlq = \critksq$, it is known that for the more general GTR models one has $\critmlq \geq \critksq$ with a strict inequality in many cases. Here, we show that this phenomenon also holds for phylogenetic reconstruction by exhibiting a family of symmetric models $Q$ and a phylogenetic reconstruction algorithm which recovers the tree from $O(\log n)$-length sequences for some branch lengths in the range $(\critksq,\critmlq)$. Second we prove that phylogenetic reconstruction under GTR models requires a polynomial sequence-length for branch lengths above $\critmlq$.

http://arxiv.org/abs/1001.3480

9830. A framework for adaptive Monte-Carlo procedures

Author(s): Bernard Lapeyre (CERMICS) and J\'er\^ome Lelong (LJK)

Abstract: Adaptive Monte Carlo methods are powerful variance reduction techniques. In this work, we propose a mathematical setting which greatly relaxes the assumptions needed by for the adaptive importance sampling techniques presented by Arouna in 2003. We establish the convergence and asymptotic normality of the adaptive Monte Carlo estimator under local assumptions which are easily verifiable in practice. We present one way of approximating the optimal importance sampling parameter using a randomly truncated stochastic algorithm. Finally, we apply this technique to the valuation of financial derivatives and our numerical experiments show that the computational time needed to achieve a given accuracy is divided by a factor up to 5.

http://arxiv.org/abs/1001.3551

9831. Solvability of general backward stochastic Volterra integral equation with non-Lipschitz coefficients

Author(s): Tianxiao Wang and Yufeng Shi

Abstract: In this paper we study the unique solvability of backward stochastic Volterra integral equations (BSVIEs in short), in terms of both the M-solutions introduced in [17] and the adapted solutions in [6], [12] or [14]. A general existence and uniqueness of M-solutions is proved under non-Lipschitz conditions by virtue of a briefer argument than the one in [17], which also extends the results in [17]. For the adapted solutions, the unique solvability of BSVIEs, under more general stochastic non-Lipschitz conditions, is obtained which generalizes the results in [6], [12] and [14].

http://arxiv.org/abs/1001.3557

9832. BSVIEs with stochastic Lipschitz coefficients and applications in finance

Author(s): Tianxiao Wang

Abstract: This paper is concerned with existence and uniqueness of M-solutions of backward stochastic Volterra integral equations (BSVIEs for short), which Lipschitz coefficients are allowed to be random, which generalize the results in [15]. Then a class of continuous time dynamic dynamic coherent risk measures is derived, allowing the riskless interest rate to be random, which is different from the case in [15].

http://arxiv.org/abs/1001.3558

9833. The measurability of hitting times

Author(s): Richard F. Bass

Abstract: Under very general conditions the hitting time of a set by a stochastic process is a stopping time. We give a new simple proof of this fact. The section theorems foroptional and predictable sets are easy corollaries of the proof.

http://arxiv.org/abs/1001.3619

9834. Schur dynamics of the Schur processes

Author(s): Alexei Borodin

Abstract: We construct discrete time Markov chains that preserve the class of Schur processes on partitions and signatures. One application is a simple exact sampling algorithm for q^{volume}-distributed skew plane partitions with an arbitrary back wall. Another application is a construction of Markov chains on infinite Gelfand-Tsetlin schemes that represent deterministic flows on the space of extreme characters of the infinite-dimensional unitary group.

http://arxiv.org/abs/1001.3442

9835. Stochastic Switching Games and Duopolistic Competition in Emissions Markets

Author(s): Michael Ludkovski

Abstract: We study optimal behavior of energy producers under a CO_2 emission abatement program. We focus on a two-player discrete-time model where each producer is sequentially optimizing her emission and production schedules. The game-theoretic aspect is captured through a reduced-form price-impact model for the CO_2 allowance price. Such duopolistic competition results in a new type of a non-zero-sum stochastic switching game on finite horizon. Existence of game Nash equilibria is established through generalization to randomized switching strategies. No uniqueness is possible and we therefore consider a variety of correlated equilibrium mechanisms. We prove existence of correlated equilibrium points in switching games and give a recursive description of equilibrium game values. A simulation-based algorithm to solve for the game values is constructed and a numerical example is presented.

http://arxiv.org/abs/1001.3455

9836. Optimal relaxed control of dissipative stochastic partial differential equations in Banach spaces

Author(s): Zdzislaw Brzezniak and Rafael Serrano

Abstract: We study an optimal relaxed control problem for a class of semilinear stochastic PDEs on Banach spaces perturbed by multiplicative noise and driven by a cylindrical Wiener process. The state equation is controlled through the nonlinear part of the drift coefficient which satisfies a dissipative-type condition with respect to the state variable. The main tools of our study are the factorization method for stochastic convolutions in UMD type-2 Banach spaces and certain compactness properties of the factorization operator and of the class of Young measures on Suslin metrisable control sets.

http://arxiv.org/abs/1001.3165

9837. Loop-Erasure of Plane Brownian Motion

Author(s): Dapeng Zhan

Abstract: We use the coupling technique to prove that there exists a loop-erasure of a plane Brownian motion stopped on exiting a simply connected domain, and the loop-erased curve is the reversal of a radial SLE$_2$ curve.

http://arxiv.org/abs/1001.3189

9838. Understanding heavy tails in a bounded world or, is a truncated heavy tail heavy or not?

Author(s): Arijit Chakrabarty and Gennady Samorodnitsky

Abstract: We address the important question of the extent to which random variables and vectors with truncated power tails retain the characteristic features of random variables and vectors with power tails. We define two truncation regimes, soft truncation regime and hard truncation regime, and show that, in the soft truncation regime, truncated power tails behave, in important respects, as if no truncation took place. On the other hand, in the hard truncation regime much of "heavy tailedness" is lost. We show how to estimate consistently the tail exponent when the tails are truncated, and suggest statistical tests to decide on whether the truncation is soft or hard. Finally, we apply our methods to two recent data sets arising from computer networks.

http://arxiv.org/abs/1001.3218

9839. Infinitely many Brownian globules with Brownian radii

Author(s): Myriam Fradon and Sylvie Roelly

Abstract: We consider an infinite system of non overlapping globules undergoing Brownian motions in R^3. The term globules means that the objects we are dealing with are spherical, but with a radius which is random and time-dependent. The dynamics is modelized by an infinite-dimensional Stochastic Differential Equation with local time. Existence and uniqueness of a strong solution is proven for such an equation with fixed deterministic initial condition. We also find a class of reversible measures.

http://arxiv.org/abs/1001.3252

9840. Regularly varying time series in Banach spaces

Author(s): Thomas Meinguet and Johan Segers

Abstract: When a spatial process is recorded over time and the observation at a given time instant is viewed as a point in a function space, the result is a time series taking values in a Banach space. To study the spatio-temporal extremal dynamics of such a time series, the latter is assumed to be jointly regularly varying. This assumption is shown to be equivalent to convergence in distribution of the rescaled time series conditionally on the event that at a given moment in time it is far away from the origin. The limit is called the tail process or the spectral process depending on the way of rescaling. These processes provide convenient starting points to study, for instance, joint survival functions, tail dependence coefficients, extremograms, extremal indices, and point processes of extremes. The theory applies to linear processes composed of infinite sums of linearly transformed independent random elements whose common distribution is regularly varying.

http://arxiv.org/abs/1001.3262

9841. Optimal stopping of expected profit and cost yields in an investment under uncertainty

Author(s): Boualem Djehiche (KTH Stockolm) and Said Hamad\`ene (LMM) and Marie Am\'elie Morlais (LMM)

Abstract: We consider a finite horizon optimal stopping problem related to trade-off strategies between expected profit and cost cash-flows of an investment under uncertainty. The optimal problem is first formulated in terms of a system of Snell envelopes for the profit and cost yields which act as obstacles to each other. We then construct both a minimal and a maximal solutions using an approximation scheme of the associated system of reflected backward SDEs. When the dependence of the cash-flows on the sources of uncertainty, such as fluctuation market prices, assumed to evolve according to a diffusion process, is made explicit, we also obtain a connection between these solutions and viscosity solutions of a system of variational inequalities (VI) with interconnected obstacles. We also provide two counter-examples showing that uniqueness of solutions of (VI) does not hold in general.

http://arxiv.org/abs/1001.3289

9842. A Milstein-type scheme without Levy area terms for SDEs driven by fractional Brownian motion

Author(s): Aur\'elien Deya (IECN) and Andreas Neuenkirch and Samy Tindel (IECN)

Abstract: In this article, we study the numerical approximation of stochastic differential equations driven by a multidimensional fractional Brownian motion (fBm) with Hurst parameter greater than 1/3. We introduce an implementable scheme for these equations, which is based on a second order Taylor expansion, where the usual Levy area terms are replaced by products of increments of the driving fBm. The convergence of our scheme is shown by means of a combination of rough paths techniques and error bounds for the discretisation of the Levy area terms.

http://arxiv.org/abs/1001.3344

9843. On a forward-backward stochastic system associated to the Burgers equation

Author(s): Ana Bela Cruzeiro and Evelina Shamarova

Abstract: We describe a probabilistic construction of $H^s$-regular solutions for the spatially periodic Burgers equation by using a characterization of this solution through a forward-backward stochastic system.

http://arxiv.org/abs/1001.3367

9844. The approach to criticality in sandpiles

Author(s): Anne Fey and Lionel Levine and and David B. Wilson

Abstract: A popular theory of self-organized criticality relates the critical behavior of driven dissipative systems to that of systems with conservation. In particular, this theory predicts that the stationary density of the abelian sandpile model should be equal to the threshold density of the corresponding fixed-energy sandpile. This "density conjecture" has been proved for the underlying graph Z. We show (by simulation or by proof) that the density conjecture is false when the underlying graph is any of Z^2, the complete graph K_n, the Cayley tree, the ladder graph, the bracelet graph, or the flower graph. Driven dissipative sandpiles continue to evolve even after a constant fraction of the sand has been lost at the sink. These results cast doubt on the validity of using fixed-energy sandpiles to explore the critical behavior of the abelian sandpile model at stationarity.

http://arxiv.org/abs/1001.3401

9845. Erasure entropies and Gibbs measures

Author(s): Aernout van Enter and Evgeny Verbitskiy

Abstract: Recently Verdu and Weissman introduced erasure entropies, which are meant to measure the information carried by one or more symbols given all of the remaining symbols in the realization of the random process or field. A natural relation to Gibbs measures has also been observed. In his short note we study this relation further, review a few earlier contributions from statistical mechanics, and provide the formula for the erasure entropy of a Gibbs measure in terms of the corresponding potentia. For some 2-dimensonal Ising models, for which Verdu and Weissman suggested a numerical procedure, we show how to obtain an exact formula for the erasure entropy. l

http://arxiv.org/abs/1001.3122

9846. Tight Bounds for Algebraic Gossip on Graphs

Author(s): Michael Borokhovich and Chen Avin and Zvi Lotker

Abstract: We study the stopping times of gossip algorithms for network coding. We analyze algebraic gossip (i.e., random linear coding) and consider three gossip algorithms for information spreading Pull, Push, and Exchange. The stopping time of algebraic gossip is known to be linear for the complete graph, but the question of determining a tight upper bound or lower bounds for general graphs is still open. We take a major step in solving this question, and prove that algebraic gossip on any graph of size n is O(D*n) where D is the maximum degree of the graph. This leads to a tight bound of Theta(n) for bounded degree graphs and an upper bound of O(n^2) for general graphs. We show that the latter bound is tight by providing an example of a graph with a stopping time of Omega(n^2). Our proofs use a novel method that relies on Jackson's queuing theorem to analyze the stopping time of network coding; this technique is likely to become useful for future research.

http://arxiv.org/abs/1001.3265

9847. Chernoff's theorem for backward propagators and applications to diffusions on manifolds

Author(s): Evelina Shamarova

Abstract: The classical Chernoff's theorem is a statement about discrete-time approximations of semigroups, where the approximations are consturcted as products of time-dependent contraction operators strongly differentiable at zero. We generalize the version of Chernoff's theorem for semigroups proved in a paper by Smolyanov et al., and obtain a theorem about descrete-time approximations of backward propagators.

http://arxiv.org/abs/1001.3373

9848. R-positivity of matrices and Hamiltonians on nearest neighbors trajectories

Author(s): Jorge Littin and Servet Martinez

Abstract: We revisit the $R-$positivity of nearest neighbors matrices on ${\ZZ_+}$ and the Gibbs measures on the set of nearest neighbors trajectories on ${\ZZ_+}$ whose Hamiltonians award either visits to sites a or visits to edges. We give conditions that guarantee the $R-$positivity or equivalently the existence of the infinite volume Gibbs measure, and we show geometrical recurrence of the associated Markov chain. In this work we generalize and sharpen results obtained in [3] and [6].

http://arxiv.org/abs/1001.2782

9849. Some properties on $G$-evaluation and its applications to $G$-martingale decomposition

Author(s): Yongsheng Song

Abstract: In this article, a sublinear expectation induced by $G$-expectation is introduced, which is called $G$-evaluation for convenience. As an application, we prove that any $\xi\in L^\beta_G(\Omega_T)$ with some $\beta>1$ the decomposition theorem holds and any $\beta>1$ integrable symmetric $G$-martingale can be represented as an It$\hat{o}'s$ integral w.r.t $G$-Brownian motion. As a byproduct, we prove a regular property for $G$-martingale: Any $G$-martingale $\{M_t\}$ has a quasi-continuous version

http://arxiv.org/abs/1001.2802

9850. The optimal control related to Riemannian manifolds and the viscosity solutions to H-J-B equations

Author(s): Xuehong Zhu

Abstract: This paper is concerned with the Dynamic Programming Principle (DPP in short) with SDEs on Riemannian manifolds. Moreover, through the DPP, we conclude that the cost function is the unique viscosity solution to the related PDEs on manifolds.

http://arxiv.org/abs/1001.2820

9851. Quantum stochastic differential equations and continuous measurements: unbounded coefficients

Author(s): Ricardo Castro Santis and Alberto Barchielli

Abstract: A natural formulation of the theory of quantum measurements in continuous time is based on quantum stochastic differential equations (Hudson-Parthasarathy equations). However, such a theory was developed only in the case of Hudson-Parthasarathy equations with bounded coefficients. By using some results on Hudson-Parthasarathy equations with unbounded coefficients, we are able to extend the theory of quantum continuous measurements to cases in which unbounded operators on the system space are involved. A significant example of a quantum optical system (the degenerate parametric oscillator) is shown to fulfill the hypotheses introduced in the general theory.

http://arxiv.org/abs/1001.2826

9852. Exact lower bounds on the exponential moments of Winsorized and truncated random variables

Author(s): Iosif Pinelis

Abstract: Exact lower bounds on the exponential moments of min(y,X) and XI{X

http://arxiv.org/abs/1001.2901

9853. Quenched effective population size

Author(s): Serik Sagitov and Peter Jagers and Vladimir Vatutin

Abstract: We study the genealogy of a geographically - or otherwise - structured version of the Wright-Fisher population model with fast migration. The new feature is that migration probabilities may change in a random fashion. Applying Takahashi's results on Markov chains with random transition matrices, we establish convergence to the Kingman coalescent, as the population size goes to infinity. This brings a novel formula for the coalescent effective population size (EPS). We call it a quenched EPS to emphasize the key feature of our model - random environment. The quenched EPS is compared with an annealed (mean-field) EPS which describes the case of constant migration probabilities obtained by averaging the random migration probabilities over possible environments.

http://arxiv.org/abs/1001.2907

9854. On refined volatility smile expansion in the Heston model

Author(s): P. Friz and S. Gerhold and A. Gulisashvili and S. Sturm

Abstract: It is known that Heston's stochastic volatility model exhibits moment explosion, and that the critical moment $s^{*}$ can be obtained by solving (numerically) a simple equation. This yields a leading order expansion for the implied volatility at large strikes: $\sigma_{BS}(k,T)^{2}T\sim \Psi (s^*-1) \times k$ (Roger Lee's moment formula). Motivated by recent "tail-wing" refinements of this moment formula, we first derive a novel tail expansion for the Heston density, sharpening previous work of Dr{\u{a}}gulescu and Yakovenko [Quant. Finance 2, 6 (2002), 443--453], and then show the validity of a refined expansion of the type $% \sigma_{BS}(k,T) ^{2}T=(\beta _{1}k^{1/2}+\beta_{2}+...)^{2}$, where all constants are explicitly known as functions of $s^*$, the Heston model parameters, spot vol and maturity $T$. In the case of the "zero-correlation" Heston model such an expansion was derived by Gulisashvili and Stein [Appl. Math. Opt., DOI: 10.1007/s002450099085]. Our methods and results may prove useful beyond the Heston model: the entire quantitative analysis is based on affine principles; at no point do we need knowledge of the (explicit, but cumbersome) closed form expression of the Fourier transform of $\log S_{T}$ (equivalently: Mellin transform of $S_{T}$% ). Secondly, our analysis reveals a new parameter ("\textit{critical slope}"% ), defined in a model free manner, which drives the second and higher order terms in tail- and implied volatility expansions.

http://arxiv.org/abs/1001.3003

9855. Stochastic differential equations with coefficients in Sobolev spaces

Author(s): Shizan Fang and Dejun Luo and Anto Thalmaier

Abstract: We consider It\^o SDE $\d X_t=\sum_{j=1}^m A_j(X_t) \d w_t^j + A_0(X_t) \d t$ on $\R^d$. The diffusion coefficients $A_1,..., A_m$ are supposed to be in the Sobolev space $W_\text{loc}^{1,p} (\R^d)$ with $p>d$, and to have linear growth; for the drift coefficient $A_0$, we consider two cases: (i) $A_0$ is continuous whose distributional divergence $\delta(A_0)$ w.r.t. the Gaussian measure $\gamma_d$ exists, (ii) $A_0$ has the Sobolev regularity $W_\text{loc}^{1,p'}$ for some $p'>1$. Assume $\int_{\R^d} \exp\big[\lambda_0\bigl(|\delta(A_0)| + \sum_{j=1}^m (|\delta(A_j)|^2 +|\nabla A_j|^2)\bigr)\big] \d\gamma_d<+\infty$ for some $\lambda_0>0$, in the case (i), if the pathwise uniqueness of solutions holds, then the push-forward $(X_t)_# \gamma_d$ admits a density with respect to $\gamma_d$. In particular, if the coefficients are bounded Lipschitz continuous, then $X_t$ leaves the Lebesgue measure $\Leb_d$ quasi-invariant. In the case (ii), we develop a method used by G. Crippa and C. De Lellis for ODE and implemented by X. Zhang for SDE, to establish the existence and uniqueness of stochastic flow of maps.

http://arxiv.org/abs/1001.3007

9856. Discrete Time and Finite State Reflected Backward Stochastic Difference Equations

Author(s): Lifen An and Shaolin Ji

Abstract: In this paper, we firstly establish the discrete time and finite state reflected backward stochastic difference equations(DF-RBSDE for short); then we explore the corresponding basic properties and theorems including the Existence and Uniqueness Theorem as well as the Comparison Theorem in our framework by "one step" method; afterwards, we show the connections between DF-RBSDE and optimal stopping time problems. For applications, we study the connection between DF-RBSDE and the general theory of g-martingales and multiple prior martingale including Doob-Mayer Decomposition Theorem and Optional Sampling Theorem in our framework; and then we apply the theory of DF-RBSDEs to multiple prior martingale and optimal stopping problems under Knightian uncertainty; finally, applying the above theories, we consider the pricing models of American Option in complete and incomplete markets.

http://arxiv.org/abs/1001.3054

9857. Incremental moments and H\"older exponents of multifractional multistable processes

Author(s): Ronan Le Gu\'evel (LMJL) and Jacques L\'evy-V\'ehel (INRIA Saclay - Ile de France)

Abstract: Multistable processes, that is, processes which are, at each ``time'', tangent to a stable process, but where the index of stability varies along the path, have been recently introduced as models for phenomena where the intensity of jumps is non constant. In this work, we give further results on (multifractional) multistable processes related to their local structure. We show that, under certain conditions, the incremental moments display a scaling behaviour, and that the pointwise exponent is, as expected, equal to the localisability index.

http://arxiv.org/abs/1001.3130

9858. Asymptotic equivalence and sufficiency for volatility estimation under microstructure noise

Author(s): Markus Rei\ss

Abstract: The basic model for high-frequency data in finance is considered, where an efficient price process is observed under microstructure noise. It is shown that this nonparametric model is in Le Cam's sense asymptotically equivalent to a Gaussian shift experiment in terms of the square root of the volatility function $\sigma$. As an application, simple rate-optimal estimators of the volatility and efficient estimators of the integrated volatility are constructed.

http://arxiv.org/abs/1001.3006

9859. Square-mean almost automorphic solutions for some stochastic differential equations

Author(s): Miaomiao Fu and Zhenxin Liu

Abstract: The concept of square-mean almost automorphy for stochastic processes is introduced. The existence and uniqueness of square-mean almost automorphic solutions to some linear and non-linear stochastic differential equations are established provided the coefficients satisfy some conditions. The asymptotic stability of the unique square-mean almost automorphic solution in square-mean sense is discussed.

http://arxiv.org/abs/1001.3049

9860. Stochastic perturbation of sweeping process and a convergence result for an associated numerical scheme

Author(s): Frederic Bernicot (LPP) and Juliette Venel (LAMAV)

Abstract: Here we present well-posedness results for first order stochastic differential inclusions, more precisely for sweeping process with a stochastic perturbation. These results are provided in combining both deterministic sweeping process theory and methods concerning the reflection of a Brownian motion. In addition, we prove convergence results for a Euler scheme, discretizing theses stochastic differential inclusions.

http://arxiv.org/abs/1001.3128

9861. A Causal Construction of Diffusion Processes

Author(s): Tadeusz Banek

Abstract: A simple nonlinear integral equation for Ito's map is obtained. Although, it does not include stochastic integrals, it does give causal construction of diffusion processes which can be easily implemented by iteration systems. Applications in financial modelling and extension to fBm are discussed.

http://arxiv.org/abs/1001.2715

9862. A Break of the Complexity of the Numerical Approximation of Nonlinear SPDEs with Multiplicative Noise

Author(s): Arnulf Jentzen and Michael Roeckner

Abstract: A new numerical method for stochastic partial differential equations (SPDEs) of evolutionary type, which is in some sense the infinite dimensional analog of Milstein's scheme for finite dimensional stochastic ordinary differential equations (SODEs), is introduced and analyzed in this article. The Milstein scheme is known to be impressively efficient for scalar one-dimensional SODEs but only for some special multidimensional SODEs due to difficult simulations of iterated stochastic integrals in the general multidimensional SODE case. It is a key observation of this article that, in contrast to what one may expect, its infinite dimensional counterpart introduced here is very easy to simulate and this, therefore, leads to a break of the complexity (number of computational operations and random variables needed to compute the scheme) in comparison to previously considered algorithms for simulating nonlinear SPDEs with multiplicative trace class noise.

http://arxiv.org/abs/1001.2751

9863. The genealogy of branching Brownian motion with absorption

Author(s): Julien Berestycki and Nathanael Berestycki and Jason Schweinsberg

Abstract: We consider a system of particles which perform branching Brownian motion with negative drift and are killed upon reaching zero, in the near-critical regime where the total population stays roughly constant with approximately N particles. We show that the characteristic time scale for the evolution of this population is of order (log N)^3, in the sense that when time is measured in these units, the scaled number of particles converges to a variant of Neveu's continuous-state branching process. Furthermore, the genealogy of the particles is then governed by a coalescent process known as the Bolthausen-Sznitman coalescent. This validates the non-rigorous predictions by Brunet, Derrida, Muller, and Munier for a closely related model.

http://arxiv.org/abs/1001.2337

9864. Continuity of a queueing integral representation in the ${M}_{\mathbf{1}}$ topology

Author(s): Guodong Pang and Ward Whitt

Abstract: We establish continuity of the integral representation $y(t)=x(t)+\int_0^th(y(s)) ds$, $t\ge0$, mapping a function $x$ into a function $y$ when the underlying function space $D$ is endowed with the Skorohod $M_1$ topology. We apply this integral representation with the continuous mapping theorem to establish heavy-traffic stochastic-process limits for many-server queueing models when the limit process has jumps unmatched in the converging processes as can occur with bursty arrival processes or service interruptions. The proof of $M_1$-continuity is based on a new characterization of the $M_1$ convergence, in which the time portions of the parametric representations are absolutely continuous with respect to Lebesgue measure, and the derivatives are uniformly bounded and converge in $L_1$.

http://arxiv.org/abs/1001.2381

9865. Branching processes in random environment which extinct at a given moment

Author(s): C. Boeinghoff and E.E. Dyakonova and G. Kersting and V.A. Vatutin

Abstract: Let ${Z_{n},n\geq 0} $ be a critical branching process in random environment and let $T$ be its moment of extinction. Under the annealed approach we prove, as $n\to \infty ,$ a limit theorem for the number of particles in the process at moment $n$ given $T=n+1$ and a functional limit theorem for the properly scaled process ${Z_{nt},\delta \leq t\leq 1-\delta} $ given $T=n+1$ and $\delta \in (0,1/2)$.

http://arxiv.org/abs/1001.2413

9866. Scaling limit of the random walk among random traps on Z^d

Author(s): Jean-Christophe Mourrat

Abstract: Attributing a positive value \tau_x to each x in Z^d, we investigate a nearest-neighbour random walk which is reversible for the measure with weights (\tau_x), often known as "Bouchaud's trap model". We assume that these weights are independent, identically distributed and non-integrable random variables (with polynomial tail), and that d > 4. We obtain the quenched subdiffusive scaling limit of the model, the limit being the fractional kinetics process. We begin our proof by expressing the random walk as a time change of a random walk among random conductances. We then focus on proving that the time change converges, under the annealed measure, to a stable subordinator. This is achieved using previous results concerning the mixing properties of the environment viewed by the time-changed random walk.

http://arxiv.org/abs/1001.2459

9867. The unscaled paths of branching Brownian motion

Author(s): Simon C. Harris and Matthew I. Roberts

Abstract: For a set $A\subset C[0,\infty)$, we give new results on the growth of the number of particles in a dyadic branching Brownian motion whose paths fall within A. We show that it is possible to work without rescaling the paths. We give large deviations probabilities as well as a more sophisticated proof of a result on growth in the number of particles along certain sets of paths. Our results reveal that the number of particles can oscillate dramatically. As a byproduct of our methods we also obtain new results on the number of particles near the frontier of the model. The methods used are entirely probabilistic.

http://arxiv.org/abs/1001.2471

9868. Fluid limit theorems for stochastic hybrid systems with application to neuron models

Author(s): K. Pakdaman and M. Thieullen and G. Wainrib

Abstract: This paper establishes limit theorems for a class of stochastic hybrid systems (continuous deterministic dynamic coupled with jump Markov processes) in the fluid limit (small jumps at high frequency), thus extending known results for jump Markov processes. We prove a functional law of large numbers with exponential convergence speed, derive a diffusion approximation and establish a functional central limit theorem. We apply these results to neuron models with stochastic ion channels, as the number of channels goes to infinity, estimating the convergence to the deterministic model. In terms of neural coding, we apply our central limit theorems to estimate numerically impact of channel noise both on frequency and spike timing coding.

http://arxiv.org/abs/1001.2474

9869. Spiders in random environment

Author(s): Christophe Gallesco and Sebastian Muller and Serguei Popov and Marina Vachkovskaia

Abstract: A spider consists of several, say $N$, particles. Particles can jump independently according to a random walk if the movement does not violate some given restriction rules. If the movement violates a rule it is not carried out. We consider random walk in random environment (RWRE) on $\Z$ as underlying random walk. We suppose the environment $\omega=(\omega_x)_{x \in \Z}$ to be elliptic, with positive drift and nestling, so that there exists a unique positive constant $\kappa$ such that $\E[((1-\omega_0)/\omega_0)^{\kappa}]=1$. The restriction rules are kept very general; we only assume transitivity and irreducibility of the spider. The main result is that the speed of a spider is positive if $\kappa/N>1$ and null if $\kappa/N<1$. In particular, if $\kappa/N <1$ a spider has null speed but the speed of a (single) RWRE is positive.

http://arxiv.org/abs/1001.2533

9870. Information Theoretic Bounds for Low-Rank Matrix Completion

Author(s): Sriram Vishwanath

Abstract: This paper studies the low-rank matrix completion problem from an information theoretic perspective. The completion problem is rephrased as a communication problem of an (uncoded) low-rank matrix source over an erasure channel. The paper then uses achievability and converse arguments to present order-wise optimal bounds for the completion problem.

http://arxiv.org/abs/1001.2331

9871. On physical diffusion and stochastic diffusion

Author(s): T. N. Narasimhan

Abstract: Although the same mathematical expression is used to describe physical diffusion and stochastic diffusion, there are intrinsic similarities and differences in their nature. A comparative study shows that characteristic terms of physical and stochastic diffusion cannot be placed exactly in one-to-one correspondence. Therefore, judgment needs to be exercised in transferring ideas between physical and stochastic diffusion.

http://arxiv.org/abs/1001.2357

9872. Asymptotics of the probability minimizing a "down-side" risk

Author(s): Hiroaki Hata and Hideo Nagai and Shuenn-Jyi Sheu

Abstract: We consider a long-term optimal investment problem where an investor tries to minimize the probability of falling below a target growth rate. From a mathematical viewpoint, this is a large deviation control problem. This problem will be shown to relate to a risk-sensitive stochastic control problem for a sufficiently large time horizon. Indeed, in our theorem we state a duality in the relation between the above two problems. Furthermore, under a multidimensional linear Gaussian model we obtain explicit solutions for the primal problem.

http://arxiv.org/abs/1001.2131

9873. Stochastic equations with boundary noise

Author(s): Roland Schnaubelt and Mark Veraar

Abstract: We study the wellposedness and pathwise regularity of semilinear non-autonomous parabolic evolution equations with boundary and interior noise in an $L^p$ setting. We obtain existence and uniqueness of mild and weak solutions. The boundary noise term is reformulated as a perturbation of a stochastic evolution equation with values in extrapolation spaces.

http://arxiv.org/abs/1001.2137

9874. Remarks on restricted Nevanlinna transforms

Author(s): Lech Jankowski and Zbigniew J. Jurek

Abstract: The Nevanlinna transform K(z), of a measure and a real constant, plays an important role in the complex analysis and more recently in the free probability theory (boolean convolution). It is shown that its restriction k(it) (the restricted Nevanlinna transform) to the imaginary axis can be expressed as the Laplace transform of the Fourier transform (characteristic function) of the corresponding measure. Finally, a relation between the Voiculescu and the boolean convolution is indicated.

http://arxiv.org/abs/1001.2154

9875. On many-server queues in heavy traffic

Author(s): Anatolii A. Puhalskii and Josh E. Reed

Abstract: We establish a heavy-traffic limit theorem on convergence in distribution for the number of customers in a many-server queue when the number of servers tends to infinity. No critical loading condition is assumed. Generally, the limit process does not have trajectories in the Skorohod space. We give conditions for the convergence to hold in the topology of compact convergence. Some new results for an infinite server are also provided.

http://arxiv.org/abs/1001.2163

9876. Consistency properties of a simulation-based estimator for dynamic processes

Author(s): Manuel S. Santos

Abstract: This paper considers a simulation-based estimator for a general class of Markovian processes and explores some strong consistency properties of the estimator. The estimation problem is defined over a continuum of invariant distributions indexed by a vector of parameters. A key step in the method of proof is to show the uniform convergence (a.s.) of a family of sample distributions over the domain of parameters. This uniform convergence holds under mild continuity and monotonicity conditions on the dynamic process. The estimator is applied to an asset pricing model with technology adoption. A challenge for this model is to generate the observed high volatility of stock markets along with the much lower volatility of other real economic aggregates.

http://arxiv.org/abs/1001.2173

9877. Convergence of some random functionals of discretized semimartingales

Author(s): Assane Diop (PMA)

Abstract: In this paper, we study the asymptotic behavior of sums of functions of the increments of a given semimartingale, taken along a regular grid whose mesh goes to 0. The function of the $i$th increment may depend on the current time, and also on the past of the semimartingale before this time. We study the convergence in probability of two types of such sums, and we also give associated central limit theorems. This extends known results when the summands are a function depending only on the increments, and this is motivated mainly by statistical applications.

http://arxiv.org/abs/1001.2182

9878. Asymptotics of q-Plancherel measures

Author(s): Valentin Feray (LaBRI) and Pierre-Lo\"ic M\'eliot (IGM-LabInfo)

Abstract: In this paper, we are interested in the asymptotic size of rows and columns of a random Young diagram under a natural deformation of the Plancherel measure coming from Hecke algebras. The first lines of such diagrams are typically of order $n$, so it does not fit in the context studied by P. Biane and P. \'Sniady. Using the theory of polynomial functions on Young diagrams of Kerov and Olshanski, we are able to compute explicitly the first- and second-order asymptotics of the length of the first rows. Our method works also for other measures, for instance those coming from Schur-Weyl representations.

http://arxiv.org/abs/1001.2180

9879. On a general many-dimensional excited random walk

Author(s): Mikhail Menshikov and Serguei Popov and Alejandro Ramirez and Marina Vachkovskaia

Abstract: The generalized excited random walk is a generalization of the excited random walk, introduced in 2003 by Benjamini and Wilson, which is a discrete-time stochastic process $(X_n,n=0,1,2,...)$ taking values on $\Z^d$, $d\geq 2$, described as follows: when the particle visits a site for the first time, it has a uniformly positive drift in a given direction $\ell$; when the particle is at a site which was already visited before, it has zero drift. Assuming uniform ellipticity and that the jumps of the process are uniformly bounded, we prove that the process is ballistic in the direction $\ell$ so that $\liminf_{n\to\infty}\frac{X_n\cdot \ell}{n}>0$. A key ingredient in the proof of this result is an estimate on the probability that the process visits less than $n^{{1/2}+\alpha}$ distinct sites by time $n$, where $\alpha$ is some positive number depending on the parameters of the model. This approach completely avoids the use of tan points and coupling methods specific to the excited random walk. Furthermore, we apply this technique to prove that the excited random walk in an i.i.d. random environment satisfies a ballistic law of large numbers and a central limit theorem.

http://arxiv.org/abs/1001.1741

9880. Functional Inequalities via Lyapunov conditions

Author(s): Patrick Cattiaux (IMT) and Arnaud Guillin

Abstract: We review here some recent results by the authors, and various coauthors, on (weak,super) Poincar\'e inequalities, transportation-information inequalities or logarithmic Sobolev inequality via a quite simple and efficient technique: Lyapunov conditions.

http://arxiv.org/abs/1001.1822

9881. Weighted Dickey-Fuller Processes for Detecting Stationarity

Author(s): Ansgar Steland

Abstract: Aiming at monitoring a time series to detect stationarity as soon as possible, we introduce monitoring procedures based on kernel-weighted sequential Dickey-Fuller (DF) processes, and related stopping times, which may be called weighted Dickey-Fuller control charts. Under rather weak assumptions, (functional) central limit theorems are established under the unit root null hypothesis and local-to-unity alternatives. For gen- eral dependent and heterogeneous innovation sequences the limit processes depend on a nuisance parameter. In this case of practical interest, one can use estimated control limits obtained from the estimated asymptotic law. Another easy-to-use approach is to transform the DF processes to obtain limit laws which are invariant with respect to the nuisance pa- rameter. We provide asymptotic theory for both approaches and compare their statistical behavior in finite samples by simulation.

http://arxiv.org/abs/1001.1833

9882. Sequentially Updated Residuals and Detection of Stationary Errors in Polynomial Regression Models

Author(s): Ansgar Steland

Abstract: The question whether a time series behaves as a random walk or as a station- ary process is an important and delicate problem, particularly arising in financial statistics, econometrics, and engineering. This paper studies the problem to detect sequentially that the error terms in a polynomial regression model no longer behave as a random walk but as a stationary process. We provide the asymptotic distribution theory for a monitoring procedure given by a control chart, i.e., a stopping time, which is related to a well known unit root test statistic calculated from sequentially updated residuals. We provide a functional central limit theorem for the corresponding stochastic process which implies a central limit theorem for the control chart. The finite sample properties are investigated by a simulation study.

http://arxiv.org/abs/1001.1845

9883. Mixing times for random k-cycles and coalescence-fragmentation chains

Author(s): Nathanael Berestycki and Oded Schramm and Ofer Zeitouni

Abstract: Let S_n be the permutation group on n elements, and consider a random walk on S_n whose step distribution is uniform on k-cycles. We prove a well-known conjecture that the mixing time of this process is (1/k) n \log n, with threshold of width linear in n. Our proofs are elementary and purely probabilistic, and do not appeal to the representation theory of S_n.

http://arxiv.org/abs/1001.1894

9884. Another observation about operator compressions

Author(s): Elizabeth S. Meckes and Mark W. Meckes

Abstract: Let $T$ be a self-adjoint operator on a finite dimensional Hilbert space. It is shown that the distribution of the eigenvalues of a compression of $T$ to a subspace of a given dimension is almost the same for almost all subspaces. This is a coordinate-free analogue of a recent result of Chatterjee and Ledoux on principal submatrices. The proof is based on measure concentration and entropy techniques, and the result improves on some aspects of the result of Chatterjee and Ledoux.

http://arxiv.org/abs/1001.1954

9885. Centering problems for probability measures on finite dimensional vector spaces

Author(s): Andrzej {\L}uczak

Abstract: The paper deals with various centering problems for probability measures on finite dimensional vector spaces. We show that for every such measure there exists a vector $h$ satisfying $\mu*\delta(h)=S(\mu*\delta (h))$ for each symmetry $S$ of $\mu$, generalizing thus Jurek's result obtained for full measures. An explicit form of the $h$ is given for infinitely divisible $\mu$. The main result of the paper consists in the analysis of quasi-decomposable (operator-semistable and operator-stable) measures and finding conditions for the existence of a `universal centering' of such a measure to a strictly quasi-decomposable one.

http://arxiv.org/abs/1001.1963

9886. Improved bounds on metastability thresholds and probabilities for generalized bootstrap percolation

Author(s): Kathrin Bringmann and Karl Mahlburg

Abstract: We generalize and improve results of Andrews, Gravner, Holroyd, Liggett, and Romik on metastability thresholds for generalized two-dimensional bootstrap percolation models, and answer several of their open problems and conjectures. Specifically, we prove slow convergence and localization bounds for Holroyd, Liggett, and Romik's k-percolation models, and in the process provide a unified and improved treatment of existing results for bootstrap, modified bootstrap, and Frobose percolation. Furthermore, we prove improved asymptotic bounds for the generating functions of partitions without k-gaps, which are also related to certain infinite probability processes relevant to these percolation models. One of our key technical probability results is also of independent interest. We prove new upper and lower bounds for the probability that a sequence of independent events with monotonically increasing probabilities contains no ``k-gap'' patterns, which interpolates the general Markov chain solution that arises in the case that all of the probabilities are equal.

http://arxiv.org/abs/1001.1977

9887. Random walks - a sequential approach

Author(s): Ansgar Steland

Abstract: In this paper sequential monitoring schemes to detect nonparametric drifts are studied for the random walk case. The procedure is based on a kernel smoother. As a by-product we obtain the asymptotics of the Nadaraya-Watson estimator and its as- sociated sequential partial sum process under non-standard sampling. The asymptotic behavior differs substantially from the stationary situation, if there is a unit root (random walk component). To obtain meaningful asymptotic results we consider local nonpara- metric alternatives for the drift component. It turns out that the rate of convergence at which the drift vanishes determines whether the asymptotic properties of the monitoring procedure are determined by a deterministic or random function. Further, we provide a theoretical result about the optimal kernel for a given alternative.

http://arxiv.org/abs/1001.1828

9888. Fractional order Taylor's series and the neo-classical inequality

Author(s): Keisuke Hara and Masanori Hino

Abstract: We prove the neo-classical inequality with the optimal constant, which was conjectured by T. J. Lyons [Rev. Mat. Iberoamericana 14 (1998) 215-310]. For the proof, we introduce the fractional order Taylor's series with residual terms. Their application to a particular function provides an identity that deduces the optimal neo-classical inequality.

http://arxiv.org/abs/1001.1775

9889. On Ergodicity, Infinite Flow and Consensus in Random Models

Author(s): Behrouz Touri and Angelia Nedi'c

Abstract: We consider the ergodicity and consensus problem for a discrete-time linear dynamic model driven by random matrices, which is equivalent to studying these concepts for the product of random matrices. Our focus is on the model where the matrices are "stochastic". We introduce a new phenomena, the infinite flow, and we study its fundamental properties and relations with the ergodicity and consensus. We establish several new and important results. The central result of this work is the infinite flow theorem establishing the role of infinite flow in the ergodicity of a general independent random model. Through the use of infinite flow, we show that the ergodicity of the model is equivalent to the ergodicity of the expected model when the matrices in the model have a common steady state in expectation and a feedback property. This result demonstrates that for such models, the expected infinite flow is both necessary and sufficient for the ergodicity. The result is providing us with a powerful deterministic characterization of the ergodicity, which renders a new elegant tool that can be used for studying the consensus and average consensus over random graphs, as well as random consensus algorithms.

http://arxiv.org/abs/1001.1890

9890. Quantum stochastic integrals as operators

Author(s): Andrzej {\L}uczak

Abstract: We construct quantum stochastic integrals for the integrator being a martingale in a von Neumann algebra, and the integrand -- a suitable process with values in the same algebra, as densely defined operators affiliated with the algebra. In the case of a finite algebra we allow the integrator to be an $L^2$--martingale in which case the integrals are $L^2$--martingales too.

http://arxiv.org/abs/1001.1959

9891. Multivariate concentration of measure type results using exchangeable pairs and size biasing

Author(s): Subhankar Ghosh

Abstract: Let $(\mathbf{W,W'})$ be an exchangeable pair of vectors in $\mathbb{R}^k$. Suppose this pair satisfies \beas E(\mathbf{W}'|\mathbf{W})=(I_k-\Lambda)\mathbf{W}+\mathbf{R(W)}. \enas If $||\mathbf{W-W'}||_2\le K$ and $\mathbf{R(W)}=0$, then concentration of measure results of following form is proved for all $\mathbf{w}\succeq 0$ when the moment generating function of $\mathbf{W}$ is finite. \beas P(\mathbf{W}\succeq\mathbf{w}),P(\mathbf{W}\preceq -\mathbf{w})\le \exp(-\frac{||\mathbf{w}||_2^2}{2K^2\nu_1}), \enas for an explicit constant $\nu_1$, where $\succeq$ stands for coordinate wise $\ge$ ordering. This result is applied to examples like complete non degenerate U-statistics. Also, we deal with the example of doubly indexed permutation statistics where $\mathbf{R(W)}\neq 0$ and obtain similar concentration of measure inequalities. Practical examples from doubly indexed permutation statistics include Mann-Whitney-Wilcoxon statistic and random intersection of two graphs. Both these two examples are used in nonparametric statistical testing. We conclude the paper with a multivariate generalization of a recent concentration result due to Ghosh and Goldstein \cite{cnm} involving bounded size bias couplings.

http://arxiv.org/abs/1001.1396

9892. Phase separation in random cluster models I: uniform upper bounds on local deviation

Author(s): Alan Hammond

Abstract: This is the first in a series of three papers that addresses the behaviour of the droplet that results, in the percolating phase, from conditioning the Fortuin-Kasteleyn random cluster model on the presence of an open dual circuit Gamma_0 encircling the origin and enclosing an area of at least (or exactly) n^2. (By the Fortuin-Kasteleyn representation, the model is a close relative of the droplet formed by conditioning the Potts model on an excess of spins of a given type.) We consider local deviation of the droplet boundary, measured in a radial sense by the maximum local roughness, MLR(Gamma_0), this being the maximum distance from a point in the circuit Gamma_0 to the boundary of the circuit's convex hull; and in a longitudinal sense by what we term maximum facet length, MFL(Gamma_0), namely, the length of the longest line segment of which the boundary of the convex hull is formed. The principal conclusion of the series of papers is the following uniform control on local deviation: that there are positive constants c and C such that the conditional probability that the normalised quantity n^{-1/3}\big(\log n \big)^{-2/3} MLR(Gamma_0) lies in the interval [c,C] tends to 1 in the high n-limit; and that the same statement holds for n^{-2/3}\big(\log n \big)^{-1/3} MFL(Gamma_0). In this way, we confirm the anticipated n^{1/3} scaling of maximum local roughness, and provide a sharp logarithmic power-law correction. This local deviation behaviour occurs by means of locally Gaussian effects constrained globally by curvature, and we believe that it arises in a range of radially defined stochastic interface models, including several in the Kardar-Parisi-Zhang universality class. This paper is devoted to proving the upper bounds in these assertions, and includes a heuristic overview of the surgical technique used in the three papers.

http://arxiv.org/abs/1001.1527

9893. Phase separation in random cluster models II: the droplet at equilibrium, and local deviation lower bounds

Author(s): Alan Hammond

Abstract: We study the droplet that results from conditioning the subcritical Fortuin-Kasteleyn random cluster model on the presence of an open circuit Gamma_0 encircling the origin and enclosing an area of at least (or exactly) n^2. We consider local deviation of the droplet boundary, measured in a radial sense by the maximum local roughness, MLR(Gamma_0), this being the maximum distance from a point in the circuit Gamma_0 to the boundary of the circuit's convex hull; and in a longitudinal sense by what we term maximum facet length, MFL(Gamma_0), namely, the length of the longest line segment of which the boundary of the convex hull is formed. We prove that that there exists a constant c > 0 such that the conditional probability that the normalised quantity n^{-1/3}\big(\log n \big)^{-2/3} MLR(Gamma_0) exceeds c tends to 1 in the high n-limit; and that the same statement holds for n^{-2/3}\big(\log n \big)^{-1/3} MFL(Gamma_0). To obtain these bounds, we exhibit the random cluster measure conditional on the presence of an open circuit trapping high area as the invariant measure of a Markov chain that resamples sections of the circuit boundary. We analyse the chain at equilibrium to prove the local roughness lower bounds. Alongside complementary upper bounds provided in arXiv:1001.1527, the fluctuations MLR(Gamma_0) and MFL(Gamma_0) are determined up to a constant factor.

http://arxiv.org/abs/1001.1528

9894. Phase separation in random cluster models III: circuit regularity

Author(s): Alan Hammond

Abstract: We study the droplet that results from conditioning the subcritical Fortuin-Kasteleyn random cluster model on the presence of an open circuit Gamma_0 encircling the origin and enclosing an area of at least (or exactly) n^2. In this paper, we prove that the resulting circuit is highly regular: we define a notion of a regeneration site in such a way that, for any such element v of Gamma_0, the circuit Gamma_0 cuts through the radial line segment through v only at v. We show that, provided that the conditioned circuit is centred at the origin in a natural sense, the set of regeneration sites reaches into all parts of the circuit, with maximal distance from one such site to the next being at most logarithmic in n with high probability. The result provides a flexible control on the conditioned circuit that permits the use of surgical techniques to bound its fluctuations, and, as such, it plays a crucial role in the derivation of bounds on the local fluctuation of the circuit carried out in arXiv:1001.1527 and arXiv:1001.1528.

http://arxiv.org/abs/1001.1529

9895. Stability of parallel queueing systems with coupled service rates

Author(s): Sem Borst and Matthieu Jonckheere and Lasse Leskel\"a

Abstract: This paper considers a parallel system of queues fed by independent arrival streams, where the service rate of each queue depends on the number of customers in all of the queues. Necessary and sufficient conditions for the stability of the system are derived, based on stochastic monotonicity and marginal drift properties of multiclass birth and death processes. These conditions yield a sharp characterization of stability for systems, where the service rate of each queue is decreasing in the number of customers in other queues, and has uniform limits as the queue lengths tend to infinity. The results are illustrated with applications where the stability region may be nonconvex.

http://arxiv.org/abs/1001.1560

9896. Critical Ising on the square lattice mixes in polynomial time

Author(s): Eyal Lubetzky and Allan Sly

Abstract: The Ising model is widely regarded as the most studied model of spin-systems in statistical physics. The focus of this paper is its dynamic (stochastic) version, the Glauber dynamics, introduced in 1963 and by now the most popular means of sampling the Ising measure. Intensive study throughout the last three decades has yielded a rigorous understanding of the spectral-gap of the dynamics on $\Z^2$ everywhere except at criticality. While the critical behavior of the Ising model has long been the focus for physicists, mathematicians have only recently developed an understanding of its critical geometry with the advent of SLE, CLE and new tools to study conformally invariant systems. A rich interplay exists between the static and dynamic models. At the static phase-transition for Ising, the dynamics is conjectured to undergo a critical slowdown: At high temperature the inverse-gap is O(1), at the critical $\beta_c$ it is polynomial in the side-length and at low temperature it is exponential in it. A seminal series of papers verified this on $\Z^2$ except at $\beta=\beta_c$ where the behavior remained a challenging open problem. Here we establish the first rigorous polynomial upper bound for the critical mixing, thus confirming the critical slowdown for the Ising model in $\Z^2$. Namely, we show that on a finite box with arbitrary (e.g. fixed, free, periodic) boundary conditions, the inverse-gap at $\beta=\beta_c$ is polynomial in the side-length. The proof harnesses recent understanding of the scaling limit of critical Fortuin-Kasteleyn representation of the Ising model together with classical tools from the analysis of Markov chains.

http://arxiv.org/abs/1001.1613

9897. Testability of minimum balanced multiway cut densities

Author(s): Marianna Bolla and Tamas Koi and Andras Kramli

Abstract: Testable weighted graph parameters and equivalent notions of testability are investigated based on papers of Laszlo Lovasz and coauthors. We prove that certain balanced minimum multiway cut densities are testable. Using this fact, quadratic programming techniques are applied to approximate some of these quantities. The problem is related to cluster analysis and statistical physics. Convergence of special noisy graph sequences is also discussed.

http://arxiv.org/abs/1001.1623

9898. Limit theorems for weakly subcritical branching processes in random environment

Author(s): V.I. Afanasyev and C. Boeinghoff and G. Kersting and V.A. Vatutin

Abstract: For a branching process in random environment it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. Interestingly there is the possibility that the process may at the same time be subcritical and, conditioned on nonextinction, 'supercritical'. This so-called weakly subcritical case is considered in this paper. We study the asymptotic survival probability and the size of the population conditioned on non-extinction. Also a functional limit theorem is proven, which makes the conditional supercriticality manifest. A main tool is a new type of functional limit theorems for conditional random walks.

http://arxiv.org/abs/1001.1672

9899. Embeddable Markov Matrices

Author(s): E B Davies

Abstract: We give an account of some results, both old and new, about any $n\times n$ Markov matrix that is embeddable in a one-parameter Markov semigroup. These include the fact that its eigenvalues must lie in a certain region in the unit ball. We prove that a well-known procedure for approximating a non-embeddable Markov matrix by an embeddable one is optimal in a certain sense.

http://arxiv.org/abs/1001.1693

9900. Relative Complexity of random walks in random sceneries

Author(s): Jon. Aaronson

Abstract: Relative complexity measures the complexity of a probability preserving transformation relative to a factor being a sequence of random variables whose exponential growth rate is the relative entropy of the extension. We prove distributional limit theorems for the relative complexity of certain zero entropy extensions: RWRSs whose associated random walks satisfy the alpha-stable CLT (alpha>1). The results give invariants for relative isomorphism of these.

http://arxiv.org/abs/1001.1433

9901. On Wiener-Hopf factors for stable processes

Author(s): Piotr Graczyk and Tomasz Jakubowski

Abstract: We give a series representation of the logarithm of the bivariate Laplace exponent $\kappa$ of $\alpha$-stable processes for almost all $\alpha \in (0,2]$.

http://arxiv.org/abs/1001.1230

9902. Fluctuations of the Longest Common Subsequence for Sequences of Independent Blocks

Author(s): Heinrich Matzinger and Felipe Torres

Abstract: The problem of the fluctuation of the Longest Common Subsequence (LCS) of two i.i.d. sequences of length $n>0$ has been open for decades. There exist contradicting conjectures on the topic. Chvatal and Sankoff conjectured in 1975 that asymptotically the order should be $n^{2/3}$, while Waterman conjectured in 1994 that asymptotically the order should be $n$. A contiguous substring consisting only of one type of symbol is called a block. In the present work, we determine the order of the fluctuation of the LCS for a special model of sequences consisting of i.i.d. blocks whose lengths are uniformly distributed on the set $\{l-1,l,l+1\}$, with $l$ a given positive integer. We showed that the fluctuation in this model is asymptotically of order $n$, which confirm Waterman's conjecture. For achieving this goal, we developed a new method which allows us to reformulate the problem of the order of the variance as a (relatively) low dimensional optimization problem.

http://arxiv.org/abs/1001.1273

9903. A functional limit theorem for partial sums of dependent random variables with infinite variance

Author(s): Bojan Basrak and Danijel Krizmani\'c and and Johan Segers

Abstract: Under an appropriate regular variation condition, the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non-Gaussian stable random variable. A functional version of this is known to be true as well, the limit process being a stable L\'evy process. The main result in the paper is that for a stationary, regularly varying sequence for which clusters of high-threshold excesses can be broken down into asymptotically independent blocks, the properly centered partial sum process still converges to a stable L\'evy process. Due to clustering, the L\'evy triple of the limit process can be different from the one in the independent case. The convergence takes place in the space of c\`adl\`ag functions endowed with Skorohod's $M_1$ topology, the more usual $J_1$ topology being inappropriate as the partial sum processes may exhibit rapid successions of jumps within temporal clusters of large values, collapsing in the limit to a single jump. The result rests on a new limit theorem for point processes which is of independent interest. The theory is applied to moving average processes, squared GARCH(1,1) processes, and stochastic volatility models.

http://arxiv.org/abs/1001.1345

9904. On some Non Asymptotic Bounds for the Euler Scheme

Author(s): Vincent Lemaire (PMA) and Stephane Menozzi (PMA)

Abstract: We obtain non asymptotic bounds for the Monte Carlo algorithm associated to the Euler discretization of some diffusion processes. The key tool is the Gaussian concentration satisfied by the density of the discretization scheme. This Gaussian concentration is derived from a Gaussian upper bound of the density of the scheme and a modification of the so-called ``Herbst argument'' used to prove Logarithmic Sobolev inequalities. We eventually establish a Gaussian lower bound for the density of the scheme that emphasizes the concentration is sharp.

http://arxiv.org/abs/1001.1347

9905. A simple reduction from a biased measure on the discrete cube to the uniform measure

Author(s): Nathan Keller

Abstract: We show that certain statements related to the Fourier-Walsh expansion of functions with respect to a biased measure on the discrete cube can be deduced from the respective results for the uniform measure by a simple reduction. In particular, we present simple generalizations to the biased measure $\mu_p$ of the Bonami-Beckner hypercontractive inequality, and of Talagrand's lower bound on the size of the boundary of subsets of the discrete cube. Our generalizations are tight up to constant factors.

http://arxiv.org/abs/1001.1167

9906. Typical Geometry, Second-Order Properties and Central Limit Theory for Iteration Stable Tessellations

Author(s): Tomasz Schreiber and Christoph Thaele

Abstract: Since the seminal work of Mecke, Nagel and Weiss, the iteration stable (STIT) tessellations have attracted considerable interest in stochastic geometry as a natural and flexible yet analytically tractable model for hierarchical spatial cell-splitting and crack-formation processes. The purpose of this paper is to describe large scale asymptotic geometry of STIT tessellations in $\mathbb{R}^d$ and more generally that of non-stationary iteration infinitely divisible tessellations. We study several aspects of the typical first-order geometry of such tessellations resorting to martingale techniques as providing a direct link between the typical characteristics of STIT tessellations and those of suitable mixtures of Poisson hyperplane tessellations. Further, we also consider second-order properties of STIT and iteration infinitely divisible tessellations, such as the variance of the total surface area of cell boundaries inside a convex observation window. Our techniques, relying on martingale theory and tools from integral geometry, allow us to give explicit and asymptotic formulae. Based on these results, we establish a functional central limit theorem for the length/surface increment processes induced by STIT tessellations. We conclude a central limit theorem for total edge length/facet surface, with normal limit distribution in the planar case and non-normal ones in all higher dimensions.

http://arxiv.org/abs/1001.0990

9907. On extrema of stable processes

Author(s): Alexey Kuznetsov

Abstract: We study Wiener-Hopf factorization and distribution of extrema for general stable processes. By connecting Wiener-Hopf factors with a certain elliptic-like function we are able to obtain many explicit and general results, such as expressions for Wiener-Hopf factors and Mellin transform of supremum in terms of double gamma functions, quasi-periodicity and functional identities for these functions, finite product representations in some special cases and identities in distribution satisfied by the supremum functional.

http://arxiv.org/abs/1001.0991

9908. Emergence of a Giant Component in Random Site Subgraphs of a d-Dimensional Hamming Torus

Author(s): David Sivakoff

Abstract: The d-dimensional Hamming torus is the graph whose vertices are all of the integer points inside an a_1 n X a_2 n X ... X a_d n box in R^d (for constants a_1, ..., a_d > 0), and whose edges connect all vertices within Hamming distance one. We study the size of the largest connected component of the subgraph generated by independently removing each vertex of the Hamming torus with probability 1-p. We show that if p=\lambda / n, then there exists \lambda_c > 0, which is the positive root of a degree d polynomial whose coefficients depend on a_1, ..., a_d, such that for \lambda < \lambda_c the largest component has O(log n) vertices (a.a.s. as n \to \infty), and for \lambda > \lambda_c the largest component has (1-q) \lambda (\prod_i a_i) n^{d-1} + o(n^{d-1}) vertices and the second largest component has O(log n) vertices (a.a.s.). An implicit formula for q < 1 is also given. Additionally, we show that if p = c log n / n, then when c < (d-1) / (\sum a_i) the site subgraph of the Hamming torus is not connected, and when c > (d-1) / (\sum a_i) the subgraph is connected (a.a.s.). We also show that the subgraph is connected precisely when it contains no isolated vertices.

http://arxiv.org/abs/1001.1007

9909. Directed polymers in random environment with heavy tails

Author(s): Antonio Auffinger and Oren Louidor

Abstract: We study the model of Directed Polymers in Random Environment in 1+1 dimensions, where the distribution at a site has a tail which decays regularly polynomially with power \alpha, where \alpha \in (0,2). After proper scaling of temperature \beta^{-1}, we show strong localization of the polymer to a favorable region in the environment where energy and entropy are best balanced. We prove that this region has a weak limit under linear scaling and identify the limiting distribution as an (\alpha, \beta)-indexed family of measures on Lipschitz curves lying inside the 45-degrees-rotated square with unit diagonal. In particular, this shows order n transversal fluctuations of the polymer. If, and only if, \alpha is small enough, we find that there exists a random critical temperature below which, but not above, the effect of the environment is macroscopic. The results carry over to d+1 dimensions for d>1 with minor modifications.

http://arxiv.org/abs/1001.1028

9910. Two-parameter Levy processes along decreasing paths

Author(s): Shai Covo (Bar Ilan University)

Abstract: Let {X_{t_1,t_2}: t_1,t_2 >= 0} be a two-parameter L\'evy process on R^d. We study basic properties of the one-parameter process {X_{x(t),y(t)}: t \in T} where x and y are, respectively, nondecreasing and nonincreasing nonnegative continuous functions on the interval T. We focus on and characterize the case where the process has stationary increments.

http://arxiv.org/abs/1001.1134

9911. Equilibrium solution to the lowest unique positive integer game

Author(s): Seung Ki Baek and Sebastian Bernhardsson

Abstract: We address the equilibrium concept of a reverse auction game so that no one can enhance the individual payoff by a unilateral change when all the others follow a certain strategy. In this approach the combinatorial possibilities to consider become very much involved even for a small number of players, which has hindered a precise analysis in previous works. We here present a systematic way to reach the solution for a general number of players, and show that this game is an example of conflict between the group and the individual interests.

http://arxiv.org/abs/1001.1065

9912. Stochastic integrals for spde's: a comparison

Author(s): Robert C. Dalang and Lluis Quer-Sardanyons

Abstract: We present the Walsh theory of stochastic integrals with respect to martingale measures, alongside of the Da Prato and Zabczyk theory of stochastic integrals with respect to Hilbert-space-valued Wiener processes and some other approaches to stochastic integration, and we explore the links between these theories. We then show how each theory can be used to study stochastic partial differential equations, with an emphasis on the stochastic heat and wave equations driven by spatially homogeneous Gaussian noise that is white in time. We compare the solutions produced by the different theories.

http://arxiv.org/abs/1001.0856

9913. Averaging over fast variables in the fluid limit for Markov chains: application to the supermarket model with memory

Author(s): M.J. Luczak and J.R. Norris

Abstract: We set out a general procedure which allows the approximation of certain Markov chains by the solutions of differential equations. The chains considered have some components which oscillate rapidly and randomly, while others are close to deterministic. The limiting dynamics are obtained by averaging the drift of the latter with respect to a local equilibrium distribution of the former. Some general estimates are proved under a uniform mixing condition on the fast variable which give explicit error probabilities for the fluid approximation. Mitzenmacher, Prabhakar and Shah \cite{MPS} introduced a variant with memory of the `join the shortest queue' or `supermarket' model, and obtained a limit picture for the case of a stable system in which the number of queues and the total arrival rate are large. In this limit, the empirical distribution of queue sizes satisfies a differential equation, while the memory of the system oscillates rapidly and randomly. We illustrate our general fluid limit estimate in giving a proof of this limit picture.

http://arxiv.org/abs/1001.0895

9914. The evolution of the cover time

Author(s): Martin T. Barlow and Jian Ding and Asaf Nachmias and Yuval Peres

Abstract: The cover time of a graph is a celebrated example of a parameter that is easy to approximate using a randomized algorithm, but for which no constant factor deterministic polynomial time approximation is known. A breakthrough due to Kahn, Kim, Lovasz and Vu yielded a (log log n)^2 polynomial time approximation. We refine this upper bound, and show that the resulting bound is sharp and explicitly computable in random graphs. Cooper and Frieze showed that the cover time of the largest component of the Erdos-Renyi random graph G(n,c/n) in the supercritical regime with c>1 fixed, is asymptotic to f(c) n \log^2 n, where f(c) tends to 1 as c tends to 1. However, our new bound implies that the cover time for the critical Erdos-Renyi random graph G(n,1/n) has order n, and shows how the cover time evolves from the critical window to the supercritical phase. Our general estimate also yields the order of the cover time for a variety of other concrete graphs, including critical percolation clusters on the Hamming hypercube {0,1}^n, on high-girth expanders, and on tori Z_n^d for fixed large d. For the graphs we consider, our results show that the blanket time, introduced by Winkler and Zuckerman, is within a constant factor of the cover time. Finally, we prove that for any connected graph, adding an edge can increase the cover time by at most a factor of 4.

http://arxiv.org/abs/1001.0609

9915. Conditional negative association for competing urns

Author(s): Jeff Kahn and Michael Neiman

Abstract: Competing urns refers to the random experiment where m balls are dropped, randomly and independently, into urns 1,...,n. Formally, we have a random map $\sigma$ from {1,...,m} to {1,...,n} with the $\sigma(i)$'s i.i.d. With $x_j$ the indicator of the event that at least $t_j$ balls land in urn j (for some threshold $t_j$), we prove conditional negative association for the random variables $x_1,...,x_n$. We mostly deal with the more general situation in which the $\sigma(i)$'s need not be identically distributed, proving results which imply conditional negative association in the i.i.d. case. Some of the results--particularly Lemma 8 on graph orientations--are thought to be of independent interest. We also give a counterexample to a negative correlation conjecture of D. Welsh, a strong version of a (still open) conjecture of G. Farr.

http://arxiv.org/abs/1001.0610

9916. A Berry Esseen Theorem for the Lightbulb Process

Author(s): Larry Goldstein and Haimeng Zhang

Abstract: In the so called lightbulb process, on days $r=1,...,n$, out of $n$ lightbulbs, all initially off, exactly $r$ bulbs, selected uniformly and independent of the past, have their status changed from off to on, or vice versa. With $X$ the number of bulbs on at the terminal time $n$, an even integer, and $\mu=n/2, \sigma^2={Var}(X)$, we have $$ \sup_{z \in \mathbb{R}} |P(\frac{X-\mu}{\sigma})-P(Z \le z)| \le \frac{n}{2\sigma^2}\Delta_0 + 1.64 \frac{n}{\sigma^3}+ \frac{2}{\sigma} $$ where $$ \Delta_0 \le \frac{1}{2\sqrt{n}} + \frac{1}{2n} + e^{-n/2} \qmq {for $n \ge 4$,} $$ yielding a bound of order $O(n^{-1/2})$ as $n \to \infty$. A similar, though slightly larger bound holds for $n$ odd. The results are shown using a version of Stein's method for bounded, monotone size bias couplings. The argument for even $n$ depends on the construction of a variable $X^s$ on the same space as $X$ which has the $X$ size bias distribution, that is, that satisfies E X g(X)=\mu Eg(X^s) \quad for all bounded continuous $g$, and for which there exists a $B \ge 0$, in this case B=2, such that $X \le X^s \le X+B$ almost surely. The argument for $n$ odd is similar to that for $n$ even, but one first couples $X$ closely to $V$, a symmetrized version of $X$, for which a size bias coupling of $V$ to $V^s$ can proceed as in the even case.

http://arxiv.org/abs/1001.0612

9917. On the distribution of the Brownian motion process on its way to hitting zero

Author(s): Konstantin Borovkov

Abstract: We present functional versions of recent results on the univariate distributions of the process $V_{x,u} = x + W_{u\tau(x)},$ $0\le u\le 1$, where $W_\bullet$ is the standard Brownian motion process, $x>0$ and $\tau (x) =\inf\{t>0 : W_{t}=-x\}$.

http://arxiv.org/abs/1001.0628

9918. Maharam extension and spectral representation of stable processes

Author(s): Emmanuel Roy (LAGA)

Abstract: We give a second look at stable processes (especially stationary) by interpreting the self-similar property at the level of the L\'evy measure as characteristic of a Maharam system. This allows us to derive structural results and their ergodic consequences. As a byproduct, we obtain a ?stable processes? proof of Banach-Lamperti Theorem for \alpha<2.

http://arxiv.org/abs/1001.0638

9919. Asymptotic properties of random matrices and pseudomatrices

Author(s): Romuald Lenczewski

Abstract: We study the asymptotics of sums of matricially free random variables called random pseudomatrices, and we compare it with that of random matrices with block-identical variances. For objects of both types we find the limit joint distributions of blocks and give their Hilbert space realizations, using operators called `matricially free Gaussian operators'. In particular, if the variance matrices are symmetric, the asymptotics of symmetric blocks of random pseudomatrices agrees with that of symmetric random blocks. We also show that blocks of random pseudomatrices are `asymptotically matricially free' whereas the corresponding symmetric random blocks are `asymptotically symmetrically matricially free', where symmetric matricial freeness is obtained from matricial freeness by an operation of symmetrization. Finally, we show that row blocks of square, lower-block-triangular and block-diagonal pseudomatrices are asymptotically free, monotone independent and boolean independent, respectively.

http://arxiv.org/abs/1001.0667

9920. On the so-called Boy or Girl Paradox

Author(s): G. D'Agostini

Abstract: A quite old problem has been recently revitalized by Leonard Mlodinow's book The Drunkard's Walk, where it is presented in a way that has definitely confused several people, that wonder why the prevalence of the name of one daughter among the population should change the probability that the other child is a girl too. I try here to discuss the problem from scratch, showing that the rarity of the name plays no role, unless the strange assumption of two identical names in the same family is taken into account. But also the name itself does not matter. What is really important is `identification', meant in an acceptation broader than usual, in the sense that a child is characterized by a set of attributes that make him/her uniquely identifiable (`that one') inside a family. The important point of how the information is acquired is also commented, suggesting an explanation of why several people tend to consider the informations "at least one boy" and "a well defined boy" (elder/youngest or of a given name) equivalent.

http://arxiv.org/abs/1001.0708

9921. A law of large numbers approximation for Markov population processes with countably many types

Author(s): A.D. Barbour and M.J. Luczak

Abstract: When modelling metapopulation dynamics, the influence of a single patch on the metapopulation depends on the number of individuals in the patch. Since the population size has no natural upper limit, this leads to systems in which there are countably infinitely many possible types of individual. Analogous considerations apply in the transmission of parasitic diseases. In this paper, we prove a law of large numbers for rather general systems of this kind, together with a rather sharp bound on the rate of convergence in an appropriately chosen weighted $\ell_1$ norm.

http://arxiv.org/abs/1001.0044

9922. Stochastic Monge-Kantorovich Problem and its Duality

Author(s): Xicheng Zhang

Abstract: In this article we prove the existence of a stochastic optimal transference plan for a stochastic Monge-Kantorovich problem by measurable selection theorem. A stochastic version of Kantorovich duality and the characterization of stochastic optimal transference plan are also established. Moreover, Wasserstein distance between two probability kernels are discussed too.

http://arxiv.org/abs/1001.0094

9923. On the generalized Feynman-Kac transformation for nearly symmetric Markov processes

Author(s): Li Ma and Wei Sun

Abstract: Suppose $X$ is a right process which is associated with a non-symmetric Dirichlet form $(\mathcal{E},D(\mathcal{E}))$ on $L^{2}(E;m)$. For $u\in D(\mathcal{E})_{e}$, we have Fukushima's decomposition: $\tilde{u}(X_{t})-\tilde{u}(X_{0})=M^{u}_{t}+N^{u}_{t}$. In this paper, we investigate the strong continuity of the generalized Feynman-Kac semigroup defined by $P^{u}_{t}f(x)=E_{x}[e^{N^{u}_{t}}f(X_{t})]$. Let $Q^{u}(f,g)=\mathcal{E}(f,g)+\mathcal{E}(u,fg)$ for $f,g\in D(\mathcal{E})_{b}$. Denote by $J_1$ the dissymmetric part of the jumping measure $J$ of $(\mathcal{E},D(\mathcal{E}))$. Under the assumption that $J_1$ is finite, we show that $(Q^{u},D(\mathcal{E})_{b})$ is lower semi-bounded if and only if there exists a constant $\alpha_0\ge 0$ such that $\|P^{u}_{t}\|_2\leq e^{\alpha_0 t}$ for every $t>0$. If one of these conditions holds, then $(P^{u}_{t})_{t\geq0}$ is strongly continuous on $L^{2}(E;m)$. If $X$ is equipped with a differential structure, then this result also holds without assuming that $J_1$ is finite.

http://arxiv.org/abs/1001.0203

9924. Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management

Author(s): Huyen Pham (PMA and Crest)

Abstract: We formulate and investigate a general stochastic control problem under a progressive enlargement of filtration. The global information is enlarged from a reference filtration and the knowledge of multiple random times together with associated marks when they occur. By working under a density hypothesis on the conditional joint distribution of the random times and marks, we prove a decomposition of the original stochastic control problem under the global filtration into classical stochastic control problems under the reference filtration, which are determined in a finite backward induction. Our method revisits and extends in particular stochastic control of diffusion processes with finite number of jumps. This study is motivated by optimization problems arising in default risk management, and we provide applications of our decomposition result for the indifference pricing of defaultable claims, and the optimal investment under bilateral counterparty risk. The solutions are expressed in terms of BSDEs involving only Brownian filtration, and remarkably without jump terms coming from the default times and marks in the global filtration.

http://arxiv.org/abs/1001.0206

9925. Bayesian nonparametric analysis for a species sampling model with finitely many types

Author(s): Annalisa Cerquetti

Abstract: We derive explicit Bayesian nonparametric analysis for a species sampling model with finitely many types of Gibbs form of type $\alpha= -1$ recently introduced in Gnedin (2009). Our results complement existing analysis under Gibbs priors of type $\alpha \in [0, 1)$ proposed in Lijoi et al. (2008). Calculations rely on a groups sequential construction of Gibbs partitions introduced in Cerquetti (2008).

http://arxiv.org/abs/1001.0245

9926. Periodically Correlated-Locally Stationary Processes

Author(s): N. Modarresi and S. Rezakhah

Abstract: In this paper we introduce a new class of non-stationary processes called, Periodically correlated-locally stationary (PC-LS) processes. It is concerned with spectral analysis of the harmonizable representation of the processes. Let $X(t)=X^s(t)+X^p(t)$ represents a stochastic process, where $X^s(t)$ is a continuous time stationary process and $X^p(t)$ is a discrete time periodically correlated (PC) process, then $X(t)$ is PC-LS. We also show that $X(t)$ is linearly correlated, which is include of periodically correlated and locally stationary (LS) processes.

http://arxiv.org/abs/1001.0296

9927. Entropy of random walk range on uniformly transient and on uniformly recurrent graphs

Author(s): David Windisch

Abstract: We study the entropy of the distribution of the set R_n of vertices visited by a simple random walk on a graph with bounded degrees in its first n steps. It is shown that this quantity grows linearly in the expected size of R_n if the graph is uniformly transient, and sublinearly in the expected size if the graph is uniformly recurrent with subexponential volume growth. This in particular answers a question asked by Benjamini, Kozma, Yadin and Yehudayoff (arXiv:0903.3179v1). In the recurrent setting, our proof shows that R_n can be compressed into a string of 0-1-bits of length sublinear in its expected size with low probability of error.

http://arxiv.org/abs/1001.0355

9928. Numerical simulation of BSDEs with drivers of quadratic growth

Author(s): Adrien Richou (IRMAR)

Abstract: We consider Markovian backward stochastic differential equations (BSDEs) with drivers of quadratic growth and bounded terminal conditions. We first show some bound estimations on the process $Z$. Then we give a new time discretization scheme for such BSDEs and we obtain an explicit convergence rate for this scheme.

http://arxiv.org/abs/1001.0401

9929. Existence and Comparisons for BSDEs in general spaces

Author(s): Samuel N. Cohen and Robert J. Elliott

Abstract: We present a theory of Backward Stochastic Differential Equations in continuous time with an arbitrary filtered probability space. No assumptions are made regarding the continuity of the filtration, or of the predictable quadratic variations of martingales in this space. We present conditions for existence and uniqueness of square-integrable solutions, using Lipschitz continuity of the driver. These conditions unite the requirements for existence in continuous and discrete time, and allow discrete processes to be embedded with continuous ones. We also present conditions for a comparison theorem, and hence construct time consistent nonlinear expectations in these general spaces.

http://arxiv.org/abs/1001.0439

9930. Intersection local times of independent fractional Brownian motions as generalized white noise functionals

Author(s): Maria Joao Oliveira and Jose Luis da Silva and and Ludwig Streit

Abstract: In this work we present expansions of intersection local times of fractional Brownian motions in $\R^d$, for any dimension $d\geq 1$, with arbitrary Hurst coefficients in $(0,1)^d$. The expansions are in terms of Wick powers of white noises (corresponding to multiple Wiener integrals), being well-defined in the sense of generalized white noise functionals. As an application of our approach, a sufficient condition on $d$ for the existence of intersection local times in $L^2$ is derived, extending the results of D. Nualart and S. Ortiz-Latorre in "Intersection Local Time for Two Independent Fractional Brownian Motions" (J. Theoret. Probab.,20(4)(2007), 759-767) to different and more general Hurst coefficients.

http://arxiv.org/abs/1001.0513

9931. Post-L1-Penalized Estimators in High-Dimensional Linear Regression Models

Author(s): Alexandre Belloni and Victor Chernozhukov

Abstract: In this paper we study the post-penalized estimator which applies ordinary, unpenalized linear regression to the model selected by the first step penalized estimators, typically the LASSO. We show that post-LASSO can perform as well or nearly as well as the LASSO in terms of the rate of convergence. We show that this performance occurs even if the LASSO-based model selection "fails", in the sense of missing some components of the "true" regression model. Furthermore, post-LASSO can perform strictly better than LASSO, in the sense of a strictly faster rate of convergence, if the LASSO-based model selection correctly includes all components of the "true" model as a subset and enough sparsity is obtained. Of course, in the extreme case, when LASSO perfectly selects the true model, the past-LASSO estimator becomes the oracle estimator. We show that the results hold in both parametric and non-parametric models; and by the "true" model we mean the best $s$-dimensional approximation to the true regression model, where the dimension $s$ is can be chosen to maximize the rate of convergence of LASSO or post-LASSO estimators. Moreover, our analysis is not limited to the LASSO estimator in the first step, and also applies to other estimators, for example, the trimmed LASSO or Dantzig selector estimator. Our analysis also highlights the importance of sparsity induced by the first estimators. That motivated us to also study the impact of trimming small components of the initial estimator to achieve a sparser support for the post-LASSO. Our analysis covers both traditional trimming, as well as a new practical completely data-driven trimming scheme that induces maximal sparsity subject to maintaining a certain goodness-of-fit.

http://arxiv.org/abs/1001.0188

9932. Asymptotic variance of random digital search trees

Author(s): Hsien-Kuei Hwang and Michael Fuchs and Vytas Zacharovas

Abstract: Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more careful normalization at the level of Poisson generating functions, which then provides an asymptotically equivalent approximation to the variance in question. Several new ingredients are also introduced such as a combined use of Laplace and Mellin transforms and a simple, mechanical technique for justifying the analytic de-Poissonization procedures involved. The methodology we develop can be easily adapted to many other problems with an underlying binomial distribution. In particular, the less expected and somewhat surprising $n(\log n)^2$-variance for certain notions of total path-length is also clarified.

http://arxiv.org/abs/1001.0095

stefano . iacus at unimi . it