Probability Abstracts 104

This document contains abstracts 6994-7235
from May-1-2008 to June-30-2008.
They have been mailed on July 8th, 2008.

6994. A Discrete Construction for Gaussian Markov Processes

Author(s): Thibaud Taillefumier

Abstract: In the L\'evy construction of Brownian motion, a Haar-derived basis of functions is used to form a finite-dimensional process $W^{N}$ and to define the Wiener process as the almost sure path-wise limit of $W^{N}$ when $N$ tends to infinity. We generalize such a construction to the class of centered Gaussian Markov processes $X$ which can be written $X_{t} = g(t) \cdot \int_{0}^{t} f(t) dW_{t}$ with $f$ and $g$ being continuous functions. We build the finite-dimensional process $X^{N}$ so that it gives an exact representation of the conditional expectation of $X$ with respect to the filtration generated by ${\lbrace X_{k/2^{N}}\rbrace}$ for $0 \leq k \leq 2^{N}$. Moreover, we prove that the process $X^{N}$ converges in distribution toward $X$.

http://arxiv.org/abs/0805.0048

6995. Optimal Robust Mean-Variance Hedging in Incomplete Financial Markets

Author(s): N. Lazrieva and T. Toronjadze

Abstract: Optimal B-robust estimate is constructed for multidimensional parameter in drift coefficient of diffusion type process with small noise. Optimal mean-variance robust (optimal V -robust) trading strategy is find to hedge in mean-variance sense the contingent claim in incomplete financial market with arbitrary information structure and misspecified volatility of asset price, which is modelled by multidimensional continuous semimartingale. Obtained results are applied to stochastic volatility model, where the model of latent volatility process contains unknown multidimensional parameter in drift coefficient and small parameter in diffusion term.

http://arxiv.org/abs/0805.0122

6996. Communication Requirements for Generating Correlated Random Variables

Author(s): Paul Cuff (Stanford University)

Abstract: Two familiar notions of correlation are rediscovered as extreme operating points for simulating a discrete memoryless channel, in which a channel output is generated based only on a description of the channel input. Wyner's "common information" coincides with the minimum description rate needed. However, when common randomness independent of the input is available, the necessary description rate reduces to Shannon's mutual information. This work characterizes the optimal tradeoff between the amount of common randomness used and the required rate of description.

http://arxiv.org/abs/0805.0065

6997. Random walks, arrangements, cell complexes, greedoids, and self-organizing libraries

Author(s): Anders Bj\"orner

Abstract: The starting point is the known fact that some much-studied random walks on permutations, such as the Tsetlin library, arise from walks on real hyperplane arrangements. This paper explores similar walks on complex hyperplane arrangements. This is achieved by involving certain cell complexes naturally associated with the arrangement. In a particular case this leads to walks on libraries with several shelves. We also show that interval greedoids give rise to random walks belonging to the same general family. Members of this family of Markov chains, based on certain semigroups, have the property that all eigenvalues of the transition matrices are non-negative real and given by a simple combinatorial formula. Background material needed for understanding the walks is reviewed in rather great detail.

http://arxiv.org/abs/0805.0083

6998. Resonances for a diffusion with small noise

Author(s): Markus Klein and Pierre-Andr\'e Zitt (MODAL'X)

Abstract: We study resonances for the generator of a diffusion with small noise in $R^d$ :$ L_\epsilon = -\epsilon\Delta + \nabla F \cdot \nabla$, when the potential F grows slowly at infinity (typically as a square root of the norm). The case when F grows fast is well known, and under suitable conditions one can show that there exists a family of exponentially small eigenvalues, related to the wells of F . We show that, for an F with a slow growth, the spectrum is R+, but we can find a family of resonances whose real parts behave as the eigenvalues of the "quick growth" case, and whose imaginary parts are small.

http://arxiv.org/abs/0805.0106

6999. Multifractal analysis in a mixed asymptotic framework

Author(s): Emmanuel Bacry and Arnaud Gloter and Marc Hoffmann and Jean-Francois Muzy

Abstract: Multifractal analysis of multiplicative random cascades is revisited within the framework of {\em mixed asymptotics}. In this new framework, statistics are estimated over a sample which size increases as the resolution scale (or the sampling period) becomes finer. This allows one to continuously interpolate between the situation where one studies a single cascade sample at arbitrary fine scales and where at fixed scale, the sample length (number of cascades realizations) becomes infinite. We show that scaling exponents of ''mixed'' partitions functions i.e., the estimator of the cumulant generating function of the cascade generator distribution, depends on some ``mixed asymptotic'' exponent $\chi$ respectively above and beyond two critical value $p_\chi^-$ and $p_\chi^+$. We study the convergence properties of partition functions in mixed asymtotics regime and establish a central limit theorem. These results are shown to remain valid within a general wavelet analysis framework. Their interpretation in terms of Besov frontier are discussed. Moreover, within the mixed asymptotic framework, we establish a ``box-counting'' multifractal formalism that can be seen as a rigorous formulation of Mandelbrot's negative dimension theory. Numerical illustrations of our purpose on specific examples are also provided.

http://arxiv.org/abs/0805.0194

7000. A Khasminskii type averaging principle for stochastic reaction-diffusion equations

Author(s): Sandra Cerrai

Abstract: We prove that an averaging principle holds for a general class of stochastic reaction-diffusion systems, having unbounded multiplicative noise, in any space dimension. We show that the classical Khasminskii approach for systems with a finite number of degrees of freedom can be extended to infinite dimensional systems.

http://arxiv.org/abs/0805.0294

7001. Averaging principle for a class of stochastic reaction-diffusion equations

Author(s): Sandra Cerrai and Mark Freidlin

Abstract: We consider the averaging principle for stochastic reaction-diffusion equations. Under some assumptions providing existence of a unique invariant measure of the fast motion with the frozen slow component, we calculate limiting slow motion. The study of solvability of Kolmogorov equations in Hilbert spaces and the analysis of regularity properties of solutions, allow to generalize the classical approach to finite-dimensional problems of this type in the case of SPDE's.

http://arxiv.org/abs/0805.0297

7002. Central Limit Theorem for a Class of Linear Systems

Author(s): Yukio Nagahata and Nobuo Yoshida

Abstract: We consider a class of interacting particle systems with values in $[0,\8)^{\zd}$, of which the binary contact path process is an example. For $d \ge 3$ and under a certain square integrability condition on the total number of the particles, we prove a central limit theorem for the density of the particles, together with upper bounds for the density of the most populated site and the replica overlap.

http://arxiv.org/abs/0805.0342

7003. A Limit Theorem for Products of Free Unitary Operators

Author(s): Vladislav Kargin

Abstract: This paper establishes necessary and sufficient conditions for the products of freely independent unitary operators to converge in distribution to the uniform law on the unit circle.

http://arxiv.org/abs/0805.0374

7004. The Player's Effect

Author(s): Ronen Gradwohl and Omer Reingold and Ariel Yadin and Amir Yehudayoff

Abstract: In a function that takes its inputs from various players, the effect of a player measures the variation he can cause in the expectation of that function. In this paper we prove a tight upper bound on the number of players with large effect, a bound that holds even when the players' inputs are only known to be pairwise independent. We also study the effect of a set of players, and show that there always exists a "small" set that, when eliminated, leaves every set with little effect. Finally, we ask whether there always exists a player with positive effect. We answer this question differently in various scenarios, depending on the properties of the function and the distribution of players' inputs. More specifically, we show that if the function is non-monotone or the distribution is only known to be pairwise independent, then it is possible that all players have 0 effect. If the distribution is pairwise independent with minimal support, on the other hand, then there must exist a player with "large" effect.

http://arxiv.org/abs/0805.0400

7005. The mixing advantage is less than 2

Author(s): Kais Hamza and Peter Jagers and Aidan Sudbury and Daniel Tokarev

Abstract: Corresponding to $n$ independent non-negative random variables $X_1,...,X_n$, are values $M_1,...,M_n$, where each $M_i$ is the expected value of the maximum of $n$ independent copies of $X_i$. We obtain an upper bound to the expected value of the maximum of $X_1,...,X_n$ in terms of $M_1,...,M_n$. This inequality is sharp in the sense that the quantity and its bound can be made as close to each other as we want. We also present related comparison results.

http://arxiv.org/abs/0805.0447

7006. Spectral gap for the interchange process in a box

Author(s): Ben Morris

Abstract: We show that the spectral gap for the interchange process (and the symmetric exclusion process) in a $d$-dimensional box of side length $L$ is asymptotic to $\pi^2/L^2$. This gives more evidence in favor of Aldous's conjecture that in any graph the spectral gap for the interchange process is the same as the spectral gap for a corresponding continuous-time random walk. Our proof uses a technique that is similar to that used by Handjani and Jungreis, who proved that Aldous's conjecture holds when the graph is a tree.

http://arxiv.org/abs/0805.0480

7007. Intermittence and nonlinear parabolic stochastic partial differential equations

Author(s): Mohammud Foondun and Davar Khoshnevisan

Abstract: We consider nonlinear parabolic SPDEs of the form $\partial_t u=\sL u + \sigma(u)\dot w$, where $\dot w$ denotes space-time white noise, $\sigma:\R\to\R$ is [globally] Lipschitz continuous, and $\sL$ is the $L^2$-generator of a L\'evy process. We present precise criteria for existence as well as uniqueness of solutions. More significantly, we prove that these solutions grow in time with at most a precise exponential rate. We establish also that when $\sigma$ is globally Lipschitz and asymptotically sublinear, the solution to the nonlinear heat equation is ``weakly intermittent,'' provided that the symmetrization of $\sL$ is recurrent and the initial data is sufficiently large. Among other things, our results lead to general formulas for the upper second-moment Liapounov exponent of the parabolic Anderson model for $\sL$ in dimension $(1+1)$. When $\sL=\kappa\partial_{xx}$ for $\kappa>0$, these formulas agree with the earlier results of statistical physics \cite{Kardar,KrugSpohn,LL63}, and also probability theory \cite{BC,CM94} in the two exactly-solvable cases where $u_0=\delta_0$ and $u_0\equiv 1$.

http://arxiv.org/abs/0805.0557

7008. A martingale approach to minimal surfaces

Author(s): Robert W. Neel

Abstract: We provide a probabilistic approach to studying minimal surfaces in three-dimensional Euclidean space. Following a discussion of the basic relationship between Brownian motion on a surface and minimality of the surface, we introduce a way of coupling Brownian motions on two minimal surfaces. This coupling is then used to study two classes of results in the theory of minimal surfaces, maximum principle-type results, such as weak and strong halfspace theorems and the maximum principle at infinity, and Liouville theorems.

http://arxiv.org/abs/0805.0556

7009. Functional moderate deviations for triangular arrays and applications

Author(s): Florence Merlevede and Magda Peligrad

Abstract: Motivated by the study of dependent random variables by coupling with independent blocks of variables, we obtain first sufficient conditions for the moderate deviation principle in its functional form for triangular arrays of independent random variables. Under some regularity assumptions our conditions are also necessary in the stationary case. The results are then applied to derive moderate deviation principles for linear processes, kernel estimators of a density and some classes of dependent random variables.

http://arxiv.org/abs/0805.0617

7010. Risk Aversion and Portfolio Selection in a Continuous-Time Model

Author(s): Jianming Xia

Abstract: The comparative statics of the optimal portfolios across individuals is carried out for a continuous-time complete market model, where the risky assets price process follows a joint geometric Brownian motion with time-dependent and deterministic coefficients. It turns out that the indirect utility functions inherit the order of risk aversion (in the Arrow-Pratt sense) from the von Neumann-Morgenstern utility functions, and therefore, a more risk-averse agent would invest less wealth (in absolute value) in the risky assets.

http://arxiv.org/abs/0805.0618

7011. Principal eigenvalue for random walk among random traps on Z^d

Author(s): Jean-Christophe Mourrat

Abstract: Let $(\tau_x)_{x \in \Z^d}$ be i.i.d. random variables with heavy (polynomial) tails. Given $a \in [0,1]$, we consider the Markov process defined by the jump rates $\omega_{x \to y} = {\tau_x}^{-(1-a)} {\tau_y}^a$ between two neighbours $x$ and $y$ in $\Z^d$. We give the asymptotic behaviour of the principal eigenvalue of the generator of this process, with Dirichlet boundary condition. The prominent feature is a phase transition that occurs at some threshold depending on the dimension. Our method relies mainly on results proved in the Appendix, which are of independent interest. They consist of a Gaussian-like upper bound on the transition kernel of any symmetric nearest-neighbour continuous-time random walk on $\Z^d$, provided its jump rates are uniformly bounded from below, together with an upper bound on the Green function when $d \ge 3$.

http://arxiv.org/abs/0805.0706

7012. Random walk weakly attracted to a wall

Author(s): Jo\"el De Coninck and Fran\c{c}ois Dunlop and Thierry Huillet

Abstract: We consider a random walk $X_n$ in $\Ze_+$, starting at $X_0=x\ge0$, with transition probabilities $$\Pe(X_{n+1}=X_n\pm1|X_n=y\ge1)={1\over2}\mp{\del\over4y+2\del}$$ and $X_{n+1}=1$ whenever $X_n=0$. We prove $\Ee X_n\sim{\rm const.} n^{1-{\del\over2}}$ as $n\nea\infty$ when $\del\in(1,2)$. The proof is based upon the Karlin-McGregor spectral representation, which is made explicit for this random walk.

http://arxiv.org/abs/0805.0729

7013. Hydrodynamic limit for a zero-range process in the Sierpinski gasket

Author(s): M. Jara

Abstract: We prove that the hydrodynamic limit of a zero-range process evolving in graphs approximating the Sierpinski gasket is given by a nonlinear heat equation. We also prove existence and uniqueness of the hydrodynamic equation by considering a finite-difference scheme.

http://arxiv.org/abs/0805.0380

7014. Levy processes and Schroedinger equation

Author(s): Nicola Cufaro Petroni and Modesto Pusterla

Abstract: We analyze the extension of the well known relation between Brownian motion and Schroedinger equation to the family of Levy processes. We propose a Levy-Schroedinger equation where the usual kinetic energy operator - the Laplacian - is generalized by means of a pseudodifferential operator whose symbol is the logarithmic characteristic of an infinitely divisible law. The Levy-Khintchin formula shows then how to write down this operator in an integro--differential form. When the underlying Levy process is stable we recover as a particular case the recently proposed fractional Schroedinger equation. A few examples are finally given and we find that there are physically relevant models (such as a form of the relativistic Schroedinger equation) that are in the domain of the possible Levy-Schroedinger equations.

http://arxiv.org/abs/0805.0503

7015. Limit theorems for additive c-free convolution

Author(s): Jiun-Chau Wang

Abstract: In this paper we find necessary and sufficient conditions for the weak convergence of c-free convolution of pairs of measures, where the measures are assumed to be infinitesimal and their support may be unbounded. These results are obtained by complex analytic methods.

http://arxiv.org/abs/0805.0607

7016. The effect of classical noise on a quantum two-level system

Author(s): Jean-Philippe Aguilar (CPT) and Nils Berglund (MAPMO)

Abstract: We consider a quantum two-level system perturbed by classical noise. The noise is implemented as a stationary diffusion process in the off-diagonal matrix elements of the Hamiltonian, representing a transverse magnetic field. We determine the invariant measure of the system and prove its uniqueness. In the case of Ornstein-Uhlenbeck noise, we determine the speed of convergence to the invariant measure. Finally, we determine an approximate one-dimensional diffusion equation for the transition probabilities. The proofs use both spectral-theoretic and probabilistic methods.

http://arxiv.org/abs/0805.0869

7017. Cut Points and Diffusions in Random Environment

Author(s): Ivan del Tenno

Abstract: In this article we investigate the asymptotic behavior of a new class of multi-dimensional diffusions in random environment. We introduce cut times in the spirit of the work done by Bolthausen, Sznitman and Zeitouni, see [4], in the discrete setting providing a decoupling effect in the process. This allows us to take advantage of an ergodic structure to derive a strong law of large numbers with possibly vanishing limiting velocity and a central limit theorem under the quenched measure.

http://arxiv.org/abs/0805.0886

7018. Behavior near the extinction time in self-similar fragmentations I: the stable case

Author(s): Christina Goldschmidt and B\'en\'edicte Haas (CEREMADE)

Abstract: The stable fragmentation with index of self-similarity $\alpha \in [-1/2,0)$ is derived by looking at the masses of the subtrees formed by discarding the parts of a $(1 + \alpha)^{-1}$--stable continuum random tree below height $t$, for $t \geq 0$. We give a detailed limiting description of the distribution of such a fragmentation, $(F(t), t \geq 0)$, as it approaches its time of extinction, $\zeta$. In particular, we show that $t^{1/\alpha}F((\zeta - t)^+)$ converges in distribution as $t \to 0$ to a non-trivial limit. In order to prove this, we go further and describe the limiting behavior of (a) an excursion of the stable height process (conditioned to have length 1) as it approaches its maximum; (b) the collection of open intervals where the excursion is above a certain level and (c) the ranked sequence of lengths of these intervals. Our principal tool is excursion theory. We also consider the last fragment to disappear and show that, with the same time and space scalings, it has a limiting distribution given in terms of a certain size-biased version of the law of $\zeta$.

http://arxiv.org/abs/0805.0967

7019. On fine properties of mixtures with respect to concentration of measure and Sobolev type inequalities

Author(s): Djalil Chafai (IMT and UPTE) and Florent Malrieu (IRMAR)

Abstract: Mixtures are convex combinations of laws. Despite this simple definition, a mixture can be far more subtle than its mixed components. For instance, mixing Gaussian laws may produce a wild potential with multiple wells. We study in the present work fine properties of mixtures with respect to concentration of measure and Gross type functional inequalities. We provide sharp Laplace bounds for Lipschitz functions in the case of generic mixtures, involving a transportation cost diameter of the mixed family. We also provide precise upper bounds for two-components mixtures. Additionally, our analysis of Gross type inequalities for two-components mixtures reveals natural relations with some kind of band isoperimetry and support constrained interpolation via mass transportation. We show that the Poincar\'e constant of a two-components mixture may remain bounded as the mixture proportion goes to 0 or 1 while the Gross constant may surprisingly blow up. Additionally, this counter-intuitive result is not reducible to support disconnections. As far as mixture of distributions are concerned, the Gross inequality is less stable than the sub-Gaussian concentration for Lipschitz functions. We illustrate our results on a gallery of concrete two-components mixtures.

http://arxiv.org/abs/0805.0987

7020. The effects of mass extinction events on the genealogy of a subdivided population

Author(s): Jesse E. Taylor and Amandine Veber

Abstract: We investigate the infinitely many demes limit of the genealogy of a sample of individuals from a subdivided population subject to sporadic mass extinction events. By exploiting a separation of timescales property of Wright's island model, we show that as the number of demes tends to infinity the limiting form of the genealogy can be described in terms of the alternation of instantaneous 'scattering' phases dominated by local demographic processes, and extended 'collecting' phases dominated by global processes. When extinction and recolonization events are local, this genealogy is given by Kingman's coalescent and the scattering phase influences only the overall rate of the process. In contrast, if the vacant demes left by a mass extinction event can be recolonized by individuals emerging from a small number of demes, then the limiting genealogy is a colaescent with simultaneous multiple mergers. In this case, the details of the within-deme population dynamics influence not only the overall rate of the coalescent process, but also the statistics of the complex mergers that can occur within sample genealogies. This study gives some insight into the genealogical consequences of mass extinction in structured populations.

http://arxiv.org/abs/0805.1010

7021. Some New Random Field Tools for Spatial Analysis

Author(s): Robert J Adler

Abstract: This is a brief review, in relatively non-technical terms, of recent advances in the theory of random field geometry. These advances have provided a collection of explicit new formulae describing mean values of a variety of geometric characteristics of excursion sets of random fields. As well as a review of the theory, we provide brief descriptions of some of the more interesting applications.

http://arxiv.org/abs/0805.1031

7022. On the Eigenspaces of Lamplighter Random Walks and Percolation Clusters on Graphs

Author(s): Franz Lehner

Abstract: We show that the Plancherel measure of the lamplighter random walk on a graph coincides with the expected spectral measure of the absorbing random walk on the Bernoulli percolation clusters. In the subcritical regime the spectrum is pure point and we construct a complete set of finitely supported eigenfunctions.

http://arxiv.org/abs/0805.0867

7023. Concentration of measure via approximated Brunn--Minkowski inequalities

Author(s): Masayoshi Watanabe

Abstract: We prove that an approximated version of the Brunn--Minkowski inequality with volume distortion coefficient implies a Gaussian concentration-of-measure phenomenon. Our main theorem is applicable to discrete spaces.

http://arxiv.org/abs/0805.0902

7024. Exactness of martingale approximation and the central limit theorem

Author(s): Dalibor Voln\'y

Abstract: The article is showing sharpness of central limit theorems of Kipnis and Varadhan, Derriennic and Lin, Maxwell and Woodroofe. In the case of the CLT of Derriennic and Lin (for Markov chains with a normal operator) it is shown that the assumption of normality cannot be relaxed. In the case of the CLT of Maxwell and Woodroofe, the example of Peligrad and Utev is improved in the sense of getting a convergence to different laws.

http://arxiv.org/abs/0805.1198

7025. The Finite Horizon Optimal Multi-Modes Switching Problem: the Viscosity Solution Approach

Author(s): Brahim El Asri and Said Hamadene

Abstract: In this paper we show existence and uniqueness of a solution for a system of m variational partial differential inequalities with inter-connected obstacles. This system is the deterministic version of the Verification Theorem of the Markovian optimal m-states switching problem. The switching cost functions are arbitrary. This problem is in relation with the valuation of firms in a financial market.

http://arxiv.org/abs/0805.1306

7026. Hydrodynamic limit of particle systems with long jumps

Author(s): M. Jara

Abstract: We consider some interacting particle processes with long-range dynamics: the zero-range and exclusion processes with long jumps. We prove that the hydrodynamic limit of these processes corresponds to a (possibly non-linear) fractional heat equation. The scaling in this case is superdiffusive. In addition, we discuss a central limit theorem for a tagged particle on the zero-range process and existence and uniqueness of solutions of the Cauchy problem for the fractional heat equation.

http://arxiv.org/abs/0805.1326

7027. Small deviations of general L\'evy processes

Author(s): Frank Aurzada and Steffen Dereich

Abstract: We study the small deviation problem $\log \mathbb{P}(\sup_{t\in[0,1]} |X_t| \leq \epsilon)$, as $\epsilon\to 0$, for general L\'evy processes $X$. The techniques enable us to determine the asymptotic rate for general real-valued L\'evy processes, which we demonstrate with many examples. As a particular consequence, we show that a L\'evy process with non-vanishing Gaussian component has the same (strong) asymptotic small deviation rate as the corresponding Brownian motion.

http://arxiv.org/abs/0805.1330

7028. A note on the enumeration of directed animals via gas considerations

Author(s): Marie Albenque

Abstract: In the literature, most of the results about the enumeration of directed animals on lattices via gas considerations are obtained by a formal passage to the limit of enumeration of directed animals on cyclical versions of the lattice. We provide here a new point of view on this phenomenon. Using the gas construction given introduced by Le Borgne and Marckert, we represent the gas process on the cyclical versions of the lattices as a cyclical Markov chain (roughly speaking, Markov chains conditioned to come back to their starting point). Then we provide a notion of convergence of graphs, such that if $(G_n)$ converges to $G$ then the gas process built on $G_n$ converges in distribution to the gas process on $G$. This gives a general tool to show that gas processes related to animals enumeration are often Markovian on some extracted line of the lattice. We provide examples and computations of new generating functions for directed animals with various sources on some families of lattices.

http://arxiv.org/abs/0805.1349

7029. Adaptive Estimation of a Distribution Function and its Density in Sup-Norm Loss by Wavelet and Spline Projections

Author(s): Evarist Gin\'e and Richard Nickl

Abstract: Given an i.i.d. sample from a distribution $F$ on $\mathbb R$ with uniformly continuous density $p_0$, purely-data driven estimators are constructed that efficiently estimate $F$ in sup-norm loss, and simultaneously estimate $p_0$ at the best possible rate of convergence over H\"{o}lder balls, also in sup-norm loss. The estimators are obtained from applying a model selection procedure close to Lepski's method with random thresholds to projections of the empirical measure onto spaces spanned by wavelets or $B$-splines. Explicit constants in the asymptotic risk of the estimator are obtained, as well as oracle-type inequalities in sup-norm loss. The random thresholds are based on suprema of Rademacher processes indexed by wavelet or spline projection kernels. This requires Bernstein-analogues of the inequalities in Koltchinskii (2006) for the deviation of suprema of empirical processes from their Rademacher symmetrizations.

http://arxiv.org/abs/0805.1404

7030. Uniform Limit Theorems for Wavelet Density Estimators

Author(s): Evarist Gin\'e and Richard Nickl

Abstract: Let $p_n (y)=\sum_k \hat \alpha_{k} \phi(y-k) + \sum_{l=0}^{j_n-1} \sum_k \hat \beta_{lk} 2^{l/2} \psi(2^ly-k)$ be the wavelet density estimator, where $\phi$, $\psi$ are a father and a mother wavelet (with compact support), $\hat \alpha_k$, $\hat \beta_{lk}$ are the empirical wavelet coefficients based on an i.i.d. sample of random variables distributed according to a density $p_0$ on $\mathbb R$, and $j_n \in \mathbb Z$, $j_n \nearrow \infty$. Several uniform limit theorems are proved: First, the almost sure rate of convergence of $\sup_{y \in \mathbb R} |p_n(y)-Ep_n(y)|$ is obtained, and a law of the logarithm for a suitably scaled version of this quantity is established. This implies that $\sup_{y \in \mathbb R} |p_n(y)-p_0(y)|$ attains the optimal almost sure rate of convergence for estimating $p_0$, if $j_n$ is suitably chosen. Second, a uniform central limit theorem as well as strong invariance principles for the distribution function of $p_n$, that is, for the stochastic processes $\sqrt n (F_n^W(s) - F(s))= \sqrt n \int_{-\infty}^s (p_n-p_0), s \in \mathbb R$, are proved; and more generally, uniform central limit theorems for the processes $\sqrt n \int (p_n-p_0)f; f \in \mathcal F$, for other Donsker classes $\mathcal F$ of interest are considered. As a statistical application, it is shown that essentially the same limit theorems can be obtained for the hard thresholding wavelet estimator introduced by Donoho, Johnstone, Kerkyacharian and Picard (1996).

http://arxiv.org/abs/0805.1406

7031. Degree-distribution stability of scale-free networks

Author(s): Zhenting Hou and Xiangxing Kong and Dinghua Shi and Guanrong Chen

Abstract: Based on the concept and techniques of first-passage probability in Markov chain theory, this letter provides a rigorous proof for the existence of the steady-state degree distribution of the scale-free network generated by the Barabasi-Albert (BA) model, and mathematically re-derives the exact analytic formulas of the distribution. The approach developed here is quite general, applicable to many other scale-free types of complex networks.

http://arxiv.org/abs/0805.1434

7032. Conditions for stochastic integrability in UMD Banach spaces

Author(s): Jan van Neerven and Mark Veraar and Lutz Weis

Abstract: A detailed theory of stochastic integration in UMD Banach spaces has been developed recently by the authors. The present paper is aimed at giving various sufficient conditions for stochastic integrability.

http://arxiv.org/abs/0805.1458

7033. Fluctuations of the partition function in the GREM with external field

Author(s): Anton Bovier and Anton Klimovsky

Abstract: We study Derrida's generalized random energy model in the presence of uniform external field. We compute the fluctuations of the ground state and of the partition function in the thermodynamic limit for all admissible values of parameters. We find that the fluctuations are described by a hierarchical structure which is obtained by a certain coarse-graining of the initial hierarchical structure of the GREM with external field. We provide an explicit formula for the free energy of the model. We also derive some large deviation results providing an expression for the free energy in a class of models with Gaussian Hamiltonians and external field. Finally, we prove that the coarse-grained parts of the system emerging in the thermodynamic limit tend to have a certain optimal magnetization, as prescribed by strength of external field and by parameters of the GREM.

http://arxiv.org/abs/0805.1478

7034. The probability of exceeding a piecewise deterministic barrier by the heavy-tailed renewal compound process

Author(s): Zbigniew Palmowski and Martijn Pistorius

Abstract: We analyze the asymptotics of crossing a high piecewise linear barriers by a renewal compound process with the subexponential jumps. The study is motivated by ruin probabilities of two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions when the initial reserves of both companies tend to infinity.

http://arxiv.org/abs/0805.1631

7035. Heat Kernel Analysis on Infinite-Dimensional Heisenberg Groups

Author(s): Bruce Driver and Maria Gordina

Abstract: We introduce a class of non-commutative Heisenberg like infinite dimensional Lie groups based on an abstract Wiener space. The Ricci curvature tensor for these groups is computed and shown to be bounded. Brownian motion and the corresponding heat kernel measures, $\{\nu_t\}_{t>0},$ are also studied. We show that these heat kernel measures admit: 1) Gaussian like upper bounds, 2) Cameron-Martin type quasi-invariance results, 3) good $L^p$ -- bounds on the corresponding Radon-Nykodim derivatives, 4) integration by parts formulas, and 5) logarithmic Sobolev inequalities. The last three results heavily rely on the boundedness of the Ricci tensor.

http://arxiv.org/abs/0805.1650

7036. Quenched and Annealed Critical Points in Polymer Pinning Models

Author(s): Kenneth S. Alexander and Nikos Zygouras

Abstract: We consider a polymer with configuration modeled by the path of a Markov chain, interacting with a potential $u+V_n$ which the chain encounters when it visits a special state 0 at time $n$. The disorder $(V_n)$ is a fixed realization of an i.i.d. sequence. The polymer is pinned, i.e. the chain spends a positive fraction of its time at state 0, when $u$ exceeds a critical value. We assume that for the Markov chain in the absence of the potential, the probability of an excursion from 0 of length $n$ has the form $n^{-c}\phi(n)$ with $c \geq 1$ and $\phi$ slowly varying. Comparing to the corresponding annealed system, in which the $V_n$ are effectively replaced by a constant, it is known that the quenched and annealed critical points differ at all temperatures for $3/22$, but only at low temperatures for $c<3/2$. For high temperatures and $3/23/2$ with arbitrary temperature we provide a new proof that the gap is positive, and extend it to $c=2$.

http://arxiv.org/abs/0805.1708

7037. On isoperimetric inequalities for log-convex measures

Author(s): Alexander V. Kolesnikov

Abstract: We study isoperimetric inequalities for measures of the type $\mu=e^{V} dx$, where $V$ is convex. Using optimal transportation techniques we estimate isoperimetric profiles for a broad class of such measures. We consider many examples and reviel some relations to the hyperbolic geometry and curvature flows.

http://arxiv.org/abs/0805.1584

7038. The covariogram determines three-dimensional convex polytopes

Author(s): Gabriele Bianchi

Abstract: The cross covariogram g_{K,L} of two convex sets K, L in R^n is the function which associates to each x in R^n the volume of the intersection of K with L+x. The problem of determining the sets from their covariogram is relevant in stochastic geometry, in probability and it is equivalent to a particular case of the phase retrieval problem in Fourier analysis. It is also relevant for the inverse problem of determining the atomic structure of a quasicrystal from its X-ray diffraction image. The two main results of this paper are that g_{K,K} determines three-dimensional convex polytopes K and that g_{K,L} determines both K and L when K and L are convex polyhedral cones satisfying certain assumptions. These results settle a conjecture of G. Matheron in the class of convex polytopes. Further results regard the known counterexamples in dimension n>=4. We also introduce and study the notion of synisothetic polytopes. This concept is related to the rearrangement of the faces of a convex polytope.

http://arxiv.org/abs/0805.1605

7039. Log-Level Comparison Principle for Small Ball Probabilities

Author(s): A. I. Nazarov

Abstract: We prove a new variant of comparison principle for logarithmic $L_2$-small ball probabilities of Gaussian processes. As an application, we obtain logarithmic small ball asymptotics for some well-known processes with smooth covariances.

http://arxiv.org/abs/0805.1773

7040. Cell contamination and branching process in random environment with immigration

Author(s): Vincent Bansaye (PMA)

Abstract: We consider a branching model for a population of dividing cells infected by parasites. Each cell receives parasites by inheritance from its mother cell and independent contamination from outside the population. Parasites multiply randomly inside the cell and are shared randomly between the two daughter cells when the cell divides. The law of the number of parasites which contaminate a given cell depends only on whether the cell is already infected or not. We determine the asymptotic behavior of the number of parasites in a cell line, which follows a branching process in random environment with state dependent immigration. We then derive a law of large numbers for the asymptotic proportions of cells with a given number of parasites. The main tools are branching processes in random environment and laws of large numbers for Markov tree.

http://arxiv.org/abs/0805.1863

7041. On the martingale probem associated to the 2D and 3D Stochastic Navier-Stokes equations

Author(s): Giuseppe Da Prato (ENS) and Arnaud Debussche (IRMAR)

Abstract: We consider the martingale problem associated to the Navier-Stokes in dimension 2 or 3. Existence is well known and it has been recently shown that markovian transition semi group associated to these equations can be constructed. We study the Kolmogorov operator associated to these equations. It can be defined formally as a differential operator on an infinite dimensional Hilbert space. It can be also defined in an abstract way as the infinitesimal generator of the transition semi group. We explicit cores for these abstract operators and identify them with the concrete differential operators on these cores. In dimension 2, the core is explicit and we can use a classical argument to prove uniqueness for the martingale problem. In dimension 3, we are only able to exhibit a core which is defined abstractly and does not allow to prove uniqueness for the martingale problem. Instead, we exhibit a core for a modified Kolmogorov operator which enables us to prove uniqueness for the martingale problem up to the time the solutions are regular.

http://arxiv.org/abs/0805.1906

7042. On one property of distances in the infinite random quadrangulation

Author(s): Maxim Krikun (IECN)

Abstract: We show that the Schaeffer's tree for an infinite quadrangulation only changes locally when changing the root of the quadrangulation. This follows from one property of distances in the infinite uniform random quadrangulation.

http://arxiv.org/abs/0805.1907

7043. Aggregation of weakly dependent doubly stochastic processes

Author(s): Lisandro J. Fermin

Abstract: The aim of this paper is to extend the aggregation convergence results given in (Dacunha-Castelle and Fermin 2005, Dacunha-Castelle and Fermin 2008) to doubly stochastic linear and nonlinear processes with weakly dependent innovations. First, we introduce a weak dependence notion for doubly stochastic processes, based in the weak dependence definition given in (Doukhan and Louhichi 1999), and we exhibe several models satisfying this notion, such as: doubly stochastic Volterra processes and doubly stochastic Bernoulli scheme with weakly dependent innovations. Afterwards we derive a central limit theorem for the partial aggregation sequence considering weakly dependent doubly stochastic processes. Finally, show a new SLLN for the covariance function of the partial aggregation process in the case of doubly stochastic Volterra processes with interactive innovations. Keywords: Aggregation, weak dependence, doubly stochastic processes, Volterra processes, Bernoulli shift, TCL, SLLN.

http://arxiv.org/abs/0805.1949

7044. On a set of transformations of Gaussian random functions

Author(s): A.I. Nazarov

Abstract: We consider a set of one-dimensional transformations of Gaussian random functions. Under natural assumptions we obtain a connection between $L_2$-small ball asymptotics of the transformed function and of the original one. Also the explicit Karhunen -- Lo\'eve expansion is obtained for a proper class of Gaussian processes.

http://arxiv.org/abs/0805.1967

7045. Stochastic calculus for convoluted L\'{e}vy processes

Author(s): Christian Bender and Tina Marquardt

Abstract: We develop a stochastic calculus for processes which are built by convoluting a pure jump, zero expectation L\'{e}vy process with a Volterra-type kernel. This class of processes contains, for example, fractional L\'{e}vy processes as studied by Marquardt [Bernoulli 12 (2006) 1090--1126.] The integral which we introduce is a Skorokhod integral. Nonetheless, we avoid the technicalities from Malliavin calculus and white noise analysis and give an elementary definition based on expectations under change of measure. As a main result, we derive an It\^{o} formula which separates the different contributions from the memory due to the convolution and from the jumps.

http://arxiv.org/abs/0805.2084

7046. Stochastic analysis on Gaussian space applied to drift estimation

Author(s): Nicolas Privault and Anthony Reveillac

Abstract: In this paper we consider the nonparametric functional estimation of the drift of Gaussian processes using Paley-Wiener and Karhunen-Lo\`eve expansions. We construct efficient estimators for the drift of such processes, and prove their minimaxity using Bayes estimators. We also construct superefficient estimators of Stein type for such drifts using the Malliavin integration by parts formula and stochastic analysis on Gaussian space, in which superharmonic functionals of the process paths play a particular role. Our results are illustrated by numerical simulations and extend the construction of James-Stein type estimators for Gaussian processes by Berger and Wolper.

http://arxiv.org/abs/0805.2002

7047. The dbar steepest descent method for orthogonal polynomials on the real line with varying weights

Author(s): K. T.-R. McLaughlin and P. D. Miller

Abstract: We obtain Plancherel-Rotach type asymptotics valid in all regions of the complex plane for orthogonal polynomials with varying weights of the form $e^{-NV(x)}$ on the real line, assuming that $V$ has only two Lipschitz continuous derivatives and that the corresponding equilibrium measure has typical support properties. As an application we extend the universality class for bulk and edge asymptotics of eigenvalue statistics in unitary invariant Hermitian random matrix theory. Our methodology involves developing a new technique of asymptotic analysis for matrix Riemann-Hilbert problems with nonanalytic jump matrices suitable for analyzing such problems even near transition points where the solution changes from oscillatory to exponential behavior.

http://arxiv.org/abs/0805.1980

7048. The Kolmogorov operator associated to a Burgers SPDE in spaces of continuous functions

Author(s): Luigi Manca

Abstract: We are concerned with a viscous Burgers equation forced by a perturbation of white noise type. We study the corresponding transition semigroup in a space of continuous functions weighted by a proper potential, and we show that the infinitesimal generator is the closure (with respect to a suitable topology) of the Kolmogorov operator associated to the stochastic equation. In the last part of the paper we use this result to solve the corresponding Fokker-Planck equation.

http://arxiv.org/abs/0805.2011

7049. Universal optimal stochastic expansions

Author(s): Simon J.A. Malham and Anke Wiese

Abstract: We study solutions to nonlinear stochastic differential systems driven by a multi-dimensional Wiener process with non-commuting diffusion vector fields, and no drift. We construct universal optimal solution expansions. They are optimal because the solution series truncated at any order is at least as accurate as the corresponding stochastic Taylor truncation in the mean-square sense. They are universal because this property is independent of the vector fields concerned. This series is the hyperbolic sine of the logarithm of the stochastic Taylor flow. Our proof utilizes the underlying Hopf algebra structure of these series, and a two-alphabet associative algebra of shuffle and concatenation operations that distinguish the coefficients of each term in the series.

http://arxiv.org/abs/0805.2340

7050. Probabilistic representation for solutions of an irregular porous media type equation

Author(s): Philippe Blanchard and Michael R\"ockner (SFB 705) and Francesco Russo (LAGA)

Abstract: We consider a porous media type equation over all of $\R^d$ with $d = 1$, with monotone discontinuous coefficients with linear growth and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. This equation is motivated by some singular behaviour arising in complex self-organized critical systems. One of the main analytic ingredients of the proof, is a new result on uniqueness of distributional solutions of a linear PDE on $\R^1$ with non-continuous coefficients.

http://arxiv.org/abs/0805.2383

7051. N/V-limit for Langevin dynamics in continuum

Author(s): Florian Conrad and Martin Grothaus

Abstract: We construct an infinite particle/infinite volume Langevin dynamics on the space of configurations in $\R^d$ having velocities as marks. The construction is done via a limiting procedure using $N$-particle dynamics in cubes $(-\lambda,\lambda]^d$ with periodic boundary conditions. A main step to this result is to derive an (improved) Ruelle bound for the canonical correlation functions of $N$-particle systems in $(-\lambda,\lambda]^d$ with periodic boundary conditions. After proving tightness of the laws of finite particle dynamics, the identification of accumulation points as martingale solutions of the Langevin equation is based on a general study of properties of measures on configuration space (and their weak limit) fulfilling a uniform Ruelle bound. Additionally, we prove that the initial/invariant distribution of the constructed dynamics is a tempered grand canonical Gibbs measure. All proofs work for general repulsive interaction potentials $\phi$ of Ruelle type (e.g. the Lennard-Jones potential) and all temperatures, densities and dimensions $d\geq 1$.

http://arxiv.org/abs/0805.2518

7052. Overcrowding and hole probabilities for random zeros on complex manifolds

Author(s): Bernard Shiffman and Steve Zelditch and Scott Zrebiec

Abstract: We give asymptotic large deviations estimates for the volume inside a domain U of the zero set of a random holomorphic section of the N-th power of a positive line bundle on a compact Kaehler manifold. In particular, we show that for all $\delta>0$, the probability that this volume differs by more than $\delta N$ from its average value is less than $\exp(-C_{\delta,U}N^{m+1})$, for some constant $C_{\delta,U}>0$. As a consequence, the "hole probability" that a random section does not vanish in U has an upper bound of the form $\exp(-C_{U}N^{m+1})$.

http://arxiv.org/abs/0805.2598

7053. Phase Transitions for the Groeth Rate of Linear Stochastic Evolutions

Author(s): Nobuo Yoshida

Abstract: We consider a simple discrete-time Markov chain with values in $[0,\infty)^{Z^d}$. The Markov chain describes various interesting examples such as oriented percolation, directed polymers in random environment, time discretizations of binary contact path process and the voter model. We study the phase transition for the growth rate of the "total number of particles" in this framework. The main results are roughly as follows: If $d \ge 3$ and the Markov chain is "not too random", then, with positive probability, the growth rate of the total number of particles is of the same order as its expectation. If on the other hand, $d=1,2$, or the Markov chain is "random enough", then the growth rate is slower than its expectation. We also discuss the above phase transition for the dual processes and its connection to the structure of invariant measures for the Markov chain with proper normalization.

http://arxiv.org/abs/0805.2652

7054. Probability theory and its models

Author(s): Paul Humphreys

Abstract: This paper argues for the status of formal probability theory as a mathematical, rather than a scientific, theory. David Freedman and Philip Stark's concept of model based probabilities is examined and is used as a bridge between the formal theory and applications.

http://arxiv.org/abs/0805.2801

7055. Dutch book in simple multivariate normal prediction: Another look

Author(s): Morris L. Eaton

Abstract: In this expository paper we describe a relatively elementary method of establishing the existence of a Dutch book in a simple multivariate normal prediction setting. The method involves deriving a nonstandard predictive distribution that is motivated by invariance. This predictive distribution satisfies an interesting identity which in turn yields an elementary demonstration of the existence of a Dutch book for a variety of possible predictive distributions.

http://arxiv.org/abs/0805.2808

7056. Generating uniform random vectors in $\QTR{bf}{Z}_{p}^{k}$: the general case

Author(s): Claudio Asci

Abstract: This paper is about the rate of convergence of the Markov chain $X_{n+1}=AX_{n}+B_{n}$ (mod $p$), where $A$ is an integer matrix with nonzero eigenvalues and ${B_{n}}_{n}$ is a sequence of independent and identically distributed integer vectors, with support not parallel to a proper subspace of $Q^{k}$ invariant under $A$. If $|\lambda_{i}|\not=1$ for all eigenvalues $\lambda_{i}$ of $A$, then $n=O((\ln p)^{2}) $ steps are sufficient and $n=O(\ln p)$ steps are necessary to have $X_{n}$ sampling from a nearly uniform distribution. Conversely, if $A$ has the eigenvalues $\lambda_{i}$ that are roots of positive integer numbers, $|\lambda_{1}|=1$ and $|\lambda_{i}|>1$ for all $i\not=1$, then $O(p^{2}) $ steps are necessary and sufficient.

http://arxiv.org/abs/0805.2830

7057. Moderate deviations for stationary sequences of Hilbert valued bounded random variables

Author(s): Sophie Dede (PMA)

Abstract: In this paper, we derive the moderate deviation principle for stationary sequences of bounded random variables with values in a Hilbert space. The conditions obtained are expressed in terms of martingale-type conditions. The main tools are martingale approximations and a new Hoeffding inequality for non adpated sequences of Hilbert-valued random variables. Applications to Cramer-Von Mises statistics, functions of linear processes and stable Markov chains are given.

http://arxiv.org/abs/0805.2899

7058. On almost randomizing channels with a short Kraus decomposition

Author(s): Guillaume Aubrun (ICJ)

Abstract: For large $d$, we study quantum channels on $\C^d$ obtained by selecting randomly $N$ independent Kraus operators according to a probability measure $\mu$ on the unitary group $\mU(d)$. When $\mu$ is the Haar measure, we show that for $N \succcurlyeq d/\e^2$, such a channel is $\e$-randomizing with high probability, which means that it maps every state within distance $\e/d$ (in operator norm) of the maximally mixed state. This slightly improves on a result by Hayden, Leung, Shor and Winter by optimizing their discretization argument. Moreover, for general $\mu$, we obtain a $\e$-randomizing channel provided $N \succcurlyeq d (\log d)^6/\e^2$. For $d=2^k$ ($k$ qubits), this includes Kraus operators obtained by tensoring $k$ random Pauli matrices. The proof uses recent results on empirical processes in Banach spaces.

http://arxiv.org/abs/0805.2900

7059. Evolution equations of the probabilistic generalization of the Voigt profile function

Author(s): Gianni Pagnini and Francesco Mainardi

Abstract: The spectrum profile that emerges in molecular spectroscopy and atmospheric radiative transfer as the combined effect of Doppler and pressure broadenings is known as the Voigt profile function. Because of its convolution integral representation, the Voigt profile can be interpreted as the probability density function of the sum of two independent random variables with Gaussian density (due to the Doppler effect) and Lorentzian density (due to the pressure effect). Since these densities belong to the class of symmetric L\'evy stable distributions, a probabilistic generalization is proposed as the convolution of two arbitrary symmetric L\'evy densities. We study the case when the widths of the considered distributions depend on a scale-factor $\tau$ that is representative of spatial inhomogeneity or temporal non-stationarity. The evolution equations for this probabilistic generalization of the Voigt function are here introduced and interpreted as generalized diffusion equations containing two Riesz space-fractional derivatives, thus classified as space-fractional diffusion equations of double order.

http://arxiv.org/abs/0711.4246

7060. Non-Markovian diffusion equations and processes: analysis and simulations

Author(s): Antonio Mura and Murad S. Taqqu and Francesco Mainardi

Abstract: In this paper we introduce and analyze a class of diffusion type equations related to certain non-Markovian stochastic processes. We start from the forward drift equation which is made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation can be interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time-evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the memory kernel K(t). We develop several applications and derive the exact solutions. We consider different stochastic models for the given equations providing path simulations.

http://arxiv.org/abs/0712.0240

7061. Disjointness of representations arising in harmonic analysis on the infinite-dimensional unitary group

Author(s): Vadim Gorin

Abstract: We prove pairwise disjointness of representations T_{z,w} of the infinite-dimensional unitary group. These representations provide a natural generalization of the regular representation for the case of "big" group U(\infty). They were introduced and studied by G.Olshanski and A.Borodin. Disjointness of the representations can be reduced to disjointness of certain probability measures on the space of paths in the Gelfand-Tsetlin graph. We prove the latter disjointness using probabilistic and combinatorial methods.

http://arxiv.org/abs/0805.2660

7062. On the distribution of the nodal sets of random spherical harmonics

Author(s): Igor Wigman

Abstract: We study the length of the nodal set of eigenfunctions of the Laplacian on the $\spheredim$-dimensional sphere. It is well known that the eigenspaces corresponding to $\eigval=n(n+\spheredim-1)$ are the spaces $\eigspc$ of spherical harmonics of degree $n$, of dimension $\eigspcdim$. We use the multiplicity of the eigenvalues to endow $\eigspc$ with the Gaussian probability measure and study the distribution of the $\spheredim$-dimensional volume of the nodal sets of a randomly chosen function. The expected volume is proportional to $\sqrt{\eigval}$. One of our main results is bounding the variance of the volume to be $O(\frac{\eigval}{\sqrt{\eigspcdim}})$. In addition to the volume of the nodal set, we study its Leray measure. For every $n$, the expected value of the Leray measure is $\frac{1}{\sqrt{2\pi}}$. We are able to determine that the asymptotic form of the variance is $\frac{const}{\eigspcdim}$.

http://arxiv.org/abs/0805.2768

7063. Generalized Backward Stochastic Differential Equation With Two Reflecting Barriers and Stochastic Quadratic Growth

Author(s): E. H. Essaky and M. Hassani

Abstract: In this paper we study one-dimensional generalized reflected backward stochastic differential equation with two barriers and stochastic quadratic growth. We prove the existence of a maximal solution when there exists a semimartingale between the barriers L and U, the generator f is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z and without assuming any P-integrability conditions on the data. The proof of our result is based on the use of a comparison theorem, an exponential transformation and an approximation technique. Our result is applied to the Dynkin game problem as well as to the American game option.

http://arxiv.org/abs/0805.2979

7064. Wigner functions and stochastically perturbed lattice dynamics

Author(s): Giada Basile (WIAS) and Stefano Olla (CEREMADE) and Herbert Spohn (D-Mutu-ZM)

Abstract: We consider lattice dynamics with a small stochastic perturbation of order \epsilon and prove that for a space-time scale of order \epsilon-1 the Wigner function evolves according to a linear transport equation describing inelastic collisions. For an energy and momentum conserving chain the transport equation predicts a slow decay, as 1/\sqrt{t}, for the energy current correlation in equilibrium. This is in agreement with previous studies using a different method.

http://arxiv.org/abs/0805.3012

7065. Asymptotics of Characteristic Polynomials of Wigner Matrices at the Edge of the Spectrum

Author(s): Holger K\"osters

Abstract: We investigate the asymptotic behaviour of the second-order correlation function of the characteristic polynomial of a Hermitian Wigner matrix at the edge of the spectrum. We show that the suitably rescaled second-order correlation function is asymptotically given by the Airy kernel, thereby generalizing the well-known result for the Gaussian Unitary Ensemble (GUE). Moreover, we obtain similar results for real-symmetric Wigner matrices.

http://arxiv.org/abs/0805.3044

7066. Convergence of dependent walks in a random scenery to fBm-local time fractional stable motions

Author(s): Serge Cohen (LSProba) and Cl\'ement Dombry (LMA)

Abstract: It is classical to approximate the distribution of fractional Brownian motion by a renormalized sum $ S_n $ of dependent Gaussian random variables. In this paper we consider such a walk $ Z_n $ that collects random rewards $ \xi_j $ for $ j \in \mathbb Z,$ when the ceiling of the walk $ S_n $ is located at $ j.$ The random reward (or scenery) $ \xi_j $ is independent of the walk and with heavy tail. We show the convergence of the sum of independent copies of $ Z_n$ suitably renormalized to a stable motion with integral representation, whose kernel is the local time of a fractional Brownian motion (fBm). This work extends a previous work where the random walk $ S_n$ had independent increments limits.

http://arxiv.org/abs/0805.3054

7067. Poisson-Dirichlet Distribution with Small Mutation Rate

Author(s): Shui Feng

Abstract: The behavior of the Poisson-Dirichlet distribution with small mutation rate is studied through large deviations. The structure of the rate function indicates that the number of alleles is finite at the instant when mutation appears. The large deviation results are then used to study the asymptotic behavior of the homozygosity, and the Poisson-Dirichlet distribution with symmetric selection. The latter shows that several alleles can coexist when selection intensity goes to infinity in a particular way as the mutation rate approaches zero.

http://arxiv.org/abs/0805.3113

7068. How T-cells use large deviations to recognize foreign antigens

Author(s): Natali Zint and Ellen Baake and Frank den Hollander

Abstract: A stochastic model for the activation of T-cells is analysed. T-cells are part of the immune system and recognize foreign antigens against a background of the body's own molecules. The model under consideration is a slight generalization of a model introduced by Van den Berg, Rand and Burroughs in 2001, and is capable of explaining how this recognition works on the basis of rare stochastic events. With the help of a refined large deviation theorem and numerical evaluation it is shown that, for a wide range of parameters, T-cells can distinguish reliably between foreign antigens and self-antigens.

http://arxiv.org/abs/q-bio/0605016

7069. Random matrices: A general approach for the least singular value problem

Author(s): Terence Tao and Van Vu

Abstract: Let $x$ be a complex random variable with mean zero and bounded variance. Let $N_{n}$ be the random matrix of size $n$ whose entries are iid copies of $x$ and $M$ be an arbitrary matrix. We give a general estimate for the least singular value of the matrix $M_{n}:=M + N_{n}$. In various special cases, our estimate extends or refines previous known results.

http://arxiv.org/abs/0805.3167

7070. Brownian Entropic Repulsion

Author(s): Itai Benjamini and Nathanael Berestycki

Abstract: We consider one-dimensional Brownian motion conditioned (in a suitable sense) to have a local time at every point and at every moment bounded by some fixed constant. Our main result shows that a phenomenon of entropic repulsion occurs: that is, this process is ballistic and has an asymptotic velocity approximately 4.58... as high as required by the conditioning (the exact value of this constant involves the first zero of a Bessel function). We also study the random walk case and show that the process is asymptotically ballistic but with an unknown speed.

http://arxiv.org/abs/0805.3326

7071. On Tightness of Mutual Dependence Upperbound for Secret-key Capacity of Multiple Terminals

Author(s): Chung Chan

Abstract: Csiszar and Narayan[3] show that the secret-key capacity with unlimited public discussion and the smallest achievable rate of communication for omniscience of a group of at least two active users sum up to the entropy rate of the discrete multiple memoryless sources for all terminals. They then derive a heuristically appealing upperbound[3,(26)] on the secret-key capacity, which is in the form of the information divergence from joint to product probability measure commonly interpreted as the mutual dependence of a set of random variables. Tightness of this bound would confirm its heuristic interpretation with the operational meaning of the secret-key capacity, i.e. the maximum mutual consensus among the active users that need not be explicitly described in public. While one can easily check that the bound is tight for any system with three or less users, testing the case with more users quickly becomes unmanageable. Yet, there is no apparent reason, other than its heuristic interpretation, that the bound is tight, nor is there a counter-example that suggests otherwise. This paper proves that the bound is indeed tight when all users are active, as a consequence of the polymatroidal structure[6] underlying the source coding problem. This already confirms the heuristic interpretation of the bound as a measure of mutual dependence of random variables. For the other case when some users are helpers, there is a counter-example with three active users and three helpers for which the bound is loose.

http://arxiv.org/abs/0805.3200

7072. Estimation in models driven by fractional Brownian motion

Author(s): Corinne Berzin and Jos\'e R. Le\'on

Abstract: Let $\{b_H(t),t\in\mathbb{R}\}$ be the fractional Brownian motion with parameter $0

http://arxiv.org/abs/0805.3394

7073. The least singular value of a random square matrix is O(n^{-1/2})

Author(s): Mark Rudelson and Roman Vershynin

Abstract: Let A be a matrix whose entries are real i.i.d. centered random variables with unit variance and suitable moment assumptions. Then the smallest singular value of A is of order n^{-1/2} with high probability. The lower estimate of this type was proved recently by the authors; in this note we establish the matching upper estimate.

http://arxiv.org/abs/0805.3407

7074. Vitesse de convergence dans le th\'{e}or\`{e}me limite central pour des cha\^{i}nes de Markov fortement ergodiques

Author(s): Lo\"ic Herv\'e

Abstract: Let $Q$ be a transition probability on a measurable space $E$ which admits an invariant probability measure, let $(X_n)_n$ be a Markov chain associated to $Q$, and let $\xi$ be a real-valued measurable function on $E$, and $S_n=\sum _{k=1}^n\xi(X_k)$. Under functional hypotheses on the action of $Q$ and the Fourier kernels $Q(t)$, we investigate the rate of convergence in the central limit theorem for the sequence $(\frac{S_n}{\sqrt{n}})_n$. According to the hypotheses, we prove that the rate is, either $\mathrm{O}(n^{-{\tau}/{2}})$ for all $\tau<1$, or $\mathrm{O}(n^{-{1}/{2}})$. We apply the spectral Nagaev's method which is improved by using a perturbation theorem of Keller and Liverani, and a majoration of $|\mathbb{E}[\mathrm{e}^{\mat hrm{i}t{S_n}/{\sqrt{n}}}]-\mathrm{e}^{{-t^2}/{2}}|$ obtained by a method of martingale difference reduction. When $E$ is not compact or $\xi$ is not bounded, the conditions required here on $Q(t)$ (in substance, some moment conditions on $\xi$) are weaker than the ones usually imposed when the standard perturbation theorem is used in the spectral method. For example, in the case of $V$-geometric ergodic chains or Lipschitz iterative models, the rate of convergence in the c.l.t. is $\mathrm{O}(n^{-{1}/{2}})$ under a third moment condition on $\xi$.

http://arxiv.org/abs/0805.3418

7075. Comparison between criteria leading to the weak invariance principle

Author(s): Olivier Durieu and Dalibor Voln\'y

Abstract: The aim of this paper is to compare various criteria leading to the central limit theorem and the weak invariance principle. These criteria are the martingale-coboundary decomposition developed by Gordin in Dokl. Akad. Nauk SSSR 188 (1969), the projective criterion introduced by Dedecker in Probab. Theory Related Fields 110 (1998), which was subsequently improved by Dedecker and Rio in Ann. Inst. H. Poincar\'{e} Probab. Statist. 36 (2000) and the condition introduced by Maxwell and Woodroofe in Ann. Probab. 28 (2000) later improved upon by Peligrad and Utev in Ann. Probab. 33 (2005). We prove that in every ergodic dynamical system with positive entropy, if we consider two of these criteria, we can find a function in $\mathbb{L}^2$ satisfying the first but not the second.

http://arxiv.org/abs/0805.3450

7076. Singular value decomposition of large random matrices (for two-way classification of microarrays)

Author(s): Marianna Bolla and Katalin Friedl and Andras Kramli

Abstract: Asymptotic behavior of the singular value decomposition (SVD) of blown up matrices and normalized blown up contingency tables exposed to Wigner-noise is investigated.It is proved that such an m\times n matrix almost surely has a constant number of large singular values (of order \sqrt{mn}), while the rest of the singular values are of order \sqrt{m+n} as m,n\to\infty. Concentration results of Alon et al. for the eigenvalues of large symmetric random matrices are adapted to the rectangular case, and on this basis, almost sure results for the singular values as well as for the corresponding isotropic subspaces are proved. An algorithm, applicable to two-way classification of microarrays, is also given that finds the underlying block structure.

http://arxiv.org/abs/0805.3476

7077. Linear Statistics of Point Processes via Orthogonal Polynomials

Author(s): E. Ryckman

Abstract: For arbitrary $\beta > 0$, we use the orthogonal polynomials techniques developed by R. Killip and I. Nenciu to study certain linear statistics associated with the circular and Jacobi $\beta$ ensembles. We identify the distribution of these statistics then prove a joint central limit theorem. In the circular case, similar statements have been proved using different methods by a number of authors. In the Jacobi case these results are new.

http://arxiv.org/abs/0805.3516

7078. Bernstein measures on convex polytopes

Author(s): Tatsuya Tate

Abstract: We define the notion of Bernstein measures and Bernstein approximations over general convex polytopes. This generalizes well-known Bernstein polynomials which are used to prove the Weierstrass approximation theorem on one dimensional intervals. We discuss some properties of Bernstein measures and approximations, and prove an asymptotic expansion of the Bernstein approximations for smooth functions which is a generalization of the asymptotic expansion of the Bernstein polynomials on the standard $m$-simplex obtained by Abel-Ivan and H\"{o}rmander. These are different from the Bergman-Bernstein approximations over Delzant polytopes recently introduced by Zelditch. We discuss relations between Bernstein approximations defined in this paper and Zelditch's Bergman-Bernstein approximations.

http://arxiv.org/abs/0805.3379

7079. Random walks in space time mixing environments

Author(s): Jean Bricmont and Antti Kupiainen

Abstract: We prove that random walks in random environments, that are exponentially mixing in space and time, are almost surely diffusive, in the sense that their scaling limit is given by the Wiener measure.

http://arxiv.org/abs/0805.3455

7080. Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model

Author(s): Federico Bassetti and Lucia Ladelli and Eugenio Regazzini

Abstract: This paper deals with a one--dimensional model for granular materials, which boils down to an inelastic version of the Kac kinetic equation, with inelasticity parameter $p>0$. In particular, the paper provides bounds for certain distances -- such as specific weighted $\chi$--distances and the Kolmogorov distance -- between the solution of that equation and the limit. It is assumed that the even part of the initial datum (which determines the asymptotic properties of the solution) belongs to the domain of normal attraction of a symmetric stable distribution with characteristic exponent $\a=2/(1+p)$. With such initial data, it turns out that the limit exists and is just the aforementioned stable distribution. A necessary condition for the relaxation to equilibrium is also proved. Some bounds are obtained without introducing any extra--condition. Sharper bounds, of an exponential type, are exhibited in the presence of additional assumptions concerning either the behaviour, near to the origin, of the initial characteristic function, or the behaviour, at infinity, of the initial probability distribution function.

http://arxiv.org/abs/0805.3508

7081. Averages of ratios of characteristic polynomials in circular beta-ensembles and super-Jack polynomials

Author(s): Sho Matsumoto

Abstract: We study the averages of ratios of characteristic polynomials over circular $\beta$-ensembles, where $\beta$ is a positive real number. Using Jack polynomial theory, we obtain three expressions for ratio averages. Two of them are given as sums of super-Jack polynomials and another one is given by a hyperdeterminant. As applications, we give duality relations for ratio averages between $\beta$ and $4/\beta$.

http://arxiv.org/abs/0805.3573

7082. On the Cluster Size Distribution for Percolation on Some General Graphs

Author(s): Antar Bandyopadhyay and Jeffrey Steif and Adam Timar

Abstract: We show that for any Cayley graph, the probability (at any $p$) that the cluster of the origin has size n decays at a well-defined exponential rate (possibly 0). For general graphs, we relate this rate being positive in the supercritical regime with the amenability/nonamenability of the underlying graph.

http://arxiv.org/abs/0805.3620

7083. Explicit error bounds for lazy reversible Markov Chain Monte Carlo

Author(s): Daniel Rudolf

Abstract: We prove explicit, i.e., non-asymptotic, error bounds for Markov Chain Monte Carlo methods, such as the Metropolis algorithm. The problem is to compute the expectation (or integral) of f with respect to a measure which can be given by a density with respect to another measure. A straight simulation of the desired distribution by a random number generator is in general not possible. Thus it is reasonable to use Markov chain sampling with a burn-in. We study such an algorithm and extend the analysis of Lovasz and Simonovits (1993) to obtain an explicit error bound.

http://arxiv.org/abs/0805.3587

7084. Success Exponent of Wiretapper: A Tradeoff between Secrecy and Reliability

Author(s): Chung Chan

Abstract: Equivocation rate has been widely used as an information-theoretic measure of security after Shannon[10]. It simplifies problems by removing the effect of atypical behavior from the system. In [9], however, Merhav and Arikan considered the alternative of using guessing exponent to analyze the Shannon's cipher system. Because guessing exponent captures the atypical behavior, the strongest expressible notion of secrecy requires the more stringent condition that the size of the key, instead of its entropy rate, to be equal to the size of the message. The relationship between equivocation and guessing exponent are also investigated in [6][7] but it is unclear which is a better measure, and whether there is a unifying measure of security. Instead of using equivocation rate or guessing exponent, we study the wiretap channel in [2] using the success exponent, defined as the exponent of a wiretapper successfully learn the secret after making an exponential number of guesses to a sequential verifier that gives yes/no answer to each guess. By extending the coding scheme in [2][5] and the converse proof in [4] with the new Overlap Lemma 5.2, we obtain a tradeoff between secrecy and reliability expressed in terms of lower bounds on the error and success exponents of authorized and respectively unauthorized decoding of the transmitted messages. From this, we obtain an inner bound to the strongly achievable public, private and guessing rate triple for which the exponents are strictly positive. The closure of this region contains the region in Theorem 1 of [2] when we treat equivocation rate as the guessing rate. It would be surprising if one can show that the subset relationship is strict, the region is tight, or a better coding scheme exists to improve it. These problems remain open.

http://arxiv.org/abs/0805.3605

7085. Multiplicative functional for reflected Brownian motion via deterministic ODE

Author(s): Krzysztof Burdzy and John M. Lee

Abstract: We prove that a sequence of semi-discrete approximations converges to a multiplicative functional for reflected Brownian motion, which intuitively represents the Lyapunov exponent for the corresponding stochastic flow. The method of proof is based on a study of the deterministic version of the problem and the excursion theory.

http://arxiv.org/abs/0805.3740

7086. Generalized Chinese restaurant construction of exchangeable Gibbs partitions and related results

Author(s): Annalisa Cerquetti

Abstract: By resorting to sequential constructions of exchangeable random partitions (Pitman, 2006), and exploiting some known facts about generalized Stirling numbers, we derive a generalized Chinese restaurant process construction of exchangeable Gibbs partitions of type $\alpha$ (Gnedin and Pitman, 2006). Our construction represents the natural theoretical probabilistic framework in which to embed some recent results about a Bayesian nonparametric treatment of estimation problems arising in genetic experiment under Gibbs, species sampling, models priors.

http://arxiv.org/abs/0805.3853

7087. Excursions away from a regular point for one-dimensional symmetric Levy processes without Gaussian part

Author(s): Kouji Yano

Abstract: The characteristic measure of excursions away from a regular point is studied for a class of symmetric Levy processes without Gaussian part. It is proved that the harmonic transform of the killed process enjoys Feller property. The result is applied to prove extremeness and oscillatory entrance properties of the excursion measure.

http://arxiv.org/abs/0805.3881

7088. A set-valued framework for birth-and-growth process

Author(s): Giacomo Aletti and Enea G. Bongiorno and Vincenzo Capasso

Abstract: We propose a set-valued framework for the well-posedness of birth-and-growth process. Our birth-and-growth model is rigorously defined as a suitable combination, involving Minkowski sum and Aumann integral, of two very general set-valued processes representing nucleation and growth respectively. The simplicity of the used geometrical approach leads us to avoid problems arising by an analytical definition of the front growth such as boundary regularities. In this framework, growth is generally anisotropic and, according to a mesoscale point of view, it is not local, i.e. for a fixed time instant, growth is the same at each space point.

http://arxiv.org/abs/0805.3912

7089. Optimal H\"older index for density states of superprocesses with (1 + \beta)-branching mechanism

Author(s): Klaus Fleischmann and Leonid Mytnik and Vitali Wachtel

Abstract: For 0 < \alpha \leq 2, a super-\alpha-stable motion X in R^d with branching of index 1 + \beta in (1,2) is considered. If d < \alpha / \beta, a dichotomy for the density of states X_t at fixed times t > 0 holds: the density function is locally H\"older continuous if d = 1 and \alpha > 1 + \beta, but locally unbounded otherwise. Moreover, in the case of continuity, we determine the optimal H\"older index.

http://arxiv.org/abs/0805.3914

7090. Optimal Investment Strategy to Minimize Occupation Time

Author(s): Erhan Bayraktar and Virginia R. Young

Abstract: We find the optimal investment strategy to minimize the expected time that an individual's wealth stays below zero, the so-called {\it occupation time}. The individual consumes at a constant rate and invests in a Black-Scholes financial market consisting of one riskless and one risky asset, with the risky asset's price process following a geometric Brownian motion. We also consider an extension of this problem by penalizing the occupation time for the degree to which wealth is negative.

http://arxiv.org/abs/0805.3981

7091. On the uniqueness of the infinite cluster of the vacant set of random interlacements

Author(s): A. Q. Teixeira

Abstract: We consider the model of random interlacements on Z^d introduced in [8]. For this model, we prove the uniqueness of the infinite component of the vacant set. As a consequence, we derive the continuity in u of the probability that the origin belongs to the infinite component of the vacant set at level u in the supercritical phase u < u_*.

http://arxiv.org/abs/0805.4106

7092. Convergence of Point Processes with Weakly Dependent Points

Author(s): Raluca Balan and Sana Louhichi

Abstract: For each $n \geq 1$, let $\{X_{j,n}\}_{1 \leq j \leq n}$ be a sequence of strictly stationary random variables. In this article, we give some asymptotic weak dependence conditions for the convergence in distribution of the point process $N_n=\sum_{j=1}^{n}\delta_{X_{j,n}}$ to an infinitely divisible point process. From the point process convergence, we obtain the convergence in distribution of the partial sum sequence $S_n=\sum_{j=1}^{n}X_{j,n}$ to an infinitely divisible random variable, whose L\'{e}vy measure is related to the canonical measure of the limiting point process. As examples, we discuss the case of triangular arrays which possess known (row-wise) dependence structures, like the strong mixing property, the association, or the dependence structure of a stochastic volatility model.

http://arxiv.org/abs/0805.4128

7093. On The Dependence Structure of Wavelet Coefficients for Spherical Random Fields

Author(s): Xiaohong Lan and Domenico Marinucci

Abstract: We consider the correlation structure of the random coefficients for a wide class of wavelet systems on the sphere which was recently introduced in the literature. We provide necessary and sufficient conditions for these coefficients to be asymptotic uncorrelated in the real and in the frequency domain. Here, the asymptotic theory is developed in the high resolution sense. Statistical applications are also discussed, in particular with reference to the analysis of cosmological data.

http://arxiv.org/abs/0805.4154

7094. Continuous time random walk and parametric subordination in fractional diffusion

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

Abstract: The well-scaled transition to the diffusion limit in the framework of the theory of continuous-time random walk (CTRW)is presented starting from its representation as an infinite series that points out the subordinated character of the CTRW itself. We treat the CTRW as a combination of a random walk on the axis of physical time with a random walk in space, both walks happening in discrete operational time. In the continuum limit we obtain a generally non-Markovian diffusion process governed by a space-time fractional diffusion equation. The essential assumption is that the probabilities for waiting times and jump-widths behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. By what we call parametric subordination, applied to a combination of a Markov process with a positively oriented L\'evy process, we generate and display sample paths for some special cases.

http://arxiv.org/abs/cond-mat/0701126

7095. Fractional Diffusion Processes: Probability Distributions and Continuous Time Random Walk

Author(s): Rudolf Gorenflo and Francesco Mainardi

Abstract: A physical-mathematical approach to anomalous diffusion may be based on fractional diffusion equations and related random walk models. The fundamental solutions of these equations can be interpreted as probability densities evolving in time of peculiar self-similar stochastic processes: an integral representation of these solutions is here presented. A more general approach to anomalous diffusion is known to be provided by the master equation for a continuous time random walk (CTRW). We show how this equation reduces to our fractional diffusion equation by a properly scaled passage to the limit of compressed waiting times and jump widths. Finally, we describe a method of simulation and display (via graphics) results of a few numerical case studies.

http://arxiv.org/abs/0709.3990

7096. Generalized Stirling permutations, families of increasing trees and urn models

Author(s): Svante Janson and Markus Kuba and Alois Panholzer

Abstract: Bona [2007+] studied the distribution of ascents, plateaux and descents in the class of Stirling permutations, introduced by Gessel and Stanley [1978]. Recently, Janson [2008+] showed the connection between Stirling permutations and plane recursive trees and proved a joint normal law for the parameters considered by Bona. Here we will consider generalized Stirling permutations extending the earlier results of Bona and Janson, and relate them with certain families of generalized plane recursive trees, and also $(k+1)$-ary increasing trees. We also give two different bijections between certain families of increasing trees, which both give as a special case a bijection between ternary increasing trees and plane recursive trees. In order to describe the (asymptotic) behaviour of the parameters of interests, we study three (generalized) Polya urn models using various methods.

http://arxiv.org/abs/0805.4084

7097. On the entropy and log-concavity of compound Poisson measures

Author(s): Oliver Johnson and Ioannis Kontoyiannis and Mokshay Madiman

Abstract: Motivated, in part, by the desire to develop an information-theoretic foundation for compound Poisson approximation limit theorems (analogous to the corresponding developments for the central limit theorem and for simple Poisson approximation), this work examines sufficient conditions under which the compound Poisson distribution has maximal entropy within a natural class of probability measures on the nonnegative integers. We show that the natural analog of the Poisson maximum entropy property remains valid if the measures under consideration are log-concave, but that it fails in general. A parallel maximum entropy result is established for the family of compound binomial measures. The proofs are largely based on ideas related to the semigroup approach introduced in recent work by Johnson for the Poisson family. Sufficient conditions are given for compound distributions to be log-concave, and specific examples are presented illustrating all the above results.

http://arxiv.org/abs/0805.4112

7098. Chains of distributions, hierarchical Bayesian models and Benford's Law

Author(s): Dennis Jang and Jung Uk Kang and Alex Kruckman and Jun Kudo and Steven J. Miller

Abstract: Alex Ely Kossovsky recently conjectured that the distribution of leading digits of a chain of probability distributions converges to Benford's law as the length of the chain grows. We prove his conjecture in many cases, and provide an interpretation in terms of products of independent random variables and a central limit theorem. An important consequence is that in hierarchical Bayesian models priors tend to satisfy Benford's Law as the number of levels of the hyper-parameters increases. We give explicit formulas for the error terms as sums of Mellin transforms, which converges extremely rapidly as the number of terms in the chain grows.

http://arxiv.org/abs/0805.4226

7099. Exact Edgeworth expansion for a L\'{e}vy process

Author(s): Heikki J. Tikanm\"aki

Abstract: The one dimensional distribution of a L\'{e}vy process is not known in general even though its characteristic function is given by the famous L\'{e}vy-Khinchine theorem. This article gives an exact series representation for the one dimensional distribution of a L\'{e}vy process satisfying certain moment conditions. Moreover, this work clarifies an old result by Cram\'{e}r on Edgeworth expansions for the distribution function of a L\'{e}vy process.

http://arxiv.org/abs/0805.4332

7100. On subexponentiality of the L\'evy measure of the diffusion inverse local time; with applications to penalizations

Author(s): Paavo Salminen and Pierre Vallois

Abstract: For a recurrent linear diffusion on $\R_+$ we study the asymptotics of the distribution of its local time at 0 as the time parameter tends to infinity. Under the assumption that the L\'evy measure of the inverse local time is subexponential this distribution behaves asymtotically as a multiple of the L\'evy measure. Using spectral representations we find the exact value of the multiple. For this we also need a result on the asymptotic behavior of the convolution of a subexponential distribution and an arbitrary distribution on $\R_+.$ The exact knowledge of the asymptotic behavior of the distribution of the local time allows us to analyze the process derived via a penalization procedure with the local time. This result generalizes the penalizations obtained in Roynette, Vallois and Yor \cite{rvyV} for Bessel processes.

http://arxiv.org/abs/0805.4353

7101. Conditioning on an Extreme Component: Model consistency and regular variation on cones

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

Abstract: Multivariate extreme value theory assumes a multivariate domain of attraction condition for the distribution of a random vector necessitating that each component satisfy a marginal domain of attraction condition. \cite{heffernan:tawn:2004} and \cite{heffernan:resnick:2007} developed an approximation to the joint distribution of the random vector by conditioning that one of the components be extreme. The prior papers left unresolved the consistency of different models obtained by conditioning on {different} components being extreme and we provide understanding of this issue. We also clarify the relationship between the conditional distributions and multivariate extreme value theory. We discuss conditions under which the two models are the same and when one can extend the conditional model to the extreme value model. We also discuss the relationship between the conditional extreme value model and standard regular variation on cones of the form $[0,\infty]\times(0,\infty]$ or $(0,\infty]\times[0,\infty]$.

http://arxiv.org/abs/0805.4373

7102. Sparse power-efficient topologies for wireless ad hoc sensor networks

Author(s): Amitabha Bagchi

Abstract: We study the problem of power-efficient routing for multihop wireless ad hoc sensor networks. The guiding insight of our work is that unlike an ad hoc wireless network, a wireless ad hoc sensor network does not require full connectivity among the nodes. As long as the sensing region is well covered by connected nodes, the network can perform its task. We consider two kinds of geometric random graphs as base interconnection structures: unit disk graphs $\UDG(2,\lambda)$ and $k$-nearest-neighbor graphs $\NN(2,k)$ built on points generated by a Poisson point process of density $\lambda$ in $\RR^2$. We provide subgraph constructions for these two models $\US(2,\lambda)$ and $\NS(2,k)$ and show that there are values $\lambda_s$ and $k_s$ above which these constructions have the following good properties: (i) they are sparse; (ii) they are power-efficient in the sense that the graph distance is no more than a constant times the Euclidean distance between any pair of points; (iii) they cover the space well; (iv) the subgraphs can be set up easily using local information at each node. We also describe a simple local algorithm for routing packets on these subgraphs.

http://arxiv.org/abs/0805.4060

7103. Queueing system with pre-scheduled random arrivals

Author(s): G. Guadagni and S. Ndreca and B. Scoppola

Abstract: We consider a point process obtained summing to each point $i$ of the set of the integer $\mathbb{Z}$ an i.i.d random variable $\xi_i$ having a variance that can be also much larger than 1. We compare the process obtained with this construction with the standard Poisson process, and we show that in some sense our process tends to converge for large variance of $\xi$ to the Poisson process in total variation. We then consider analytically and numerically a simple queueing system having our process as arrival process. This model is motivated by the study of air traffic systems.

http://arxiv.org/abs/0805.4472

7104. Random Walks on Discrete Cylinders and Random Interlacements

Author(s): Alain-Sol Sznitman

Abstract: We explore some of the connections between the local picture left by the trace of simple random walk on a discrete cylinder with base a d-dimensional torus, d at least 2, of side-length N running for times of order N^{2d} and the model of random interlacements recently introduced in arXiv:0704.2560. In particular we show that when the base becomes large, in the neighborhood of a point of the cylinder with a vertical component of order N^d, the complement of the set of points visited by the walk up to times of order N^{2d}, is close in distribution to the law of the vacant set of random interlacements at a level which is determined by an independent Brownian local time. The limit of the local pictures in the neighborhood of finitely many points is also derived.

http://arxiv.org/abs/0805.4516

7105. On Upper Bounds for the Tail Distribution of Geometric Sums of Subexponential Random Variables

Author(s): Andrew Richards

Abstract: The approach used by Kalashnikov and Tsitsiashvili for constructing upper bounds for the tail distribution of a geometric sum with subexponential summands is reconsidered. By expressing the problem in a more probabilistic light, several improvements and one correction are made, which enables the constructed bound to be significantly tighter. Several examples are given, showing how to implement the theoretical result.

http://arxiv.org/abs/0805.4548

7106. Asymptotic Properties of an Estimator of the Drift Coefficients of Multidimensional Ornstein-Uhlenbeck Processes that are not Necessarily Stable

Author(s): Gopal K. Basak and Philip Lee

Abstract: In this paper, we investigate the consistency and asymptotic efficiency of an estimator of the drift matrix, $F$, of Ornstein-Uhlenbeck processes that are not necessarily stable. We consider all the cases. (1) The eigenvalues of $F$ are in the right half space (i.e., eigenvalues with positive real parts). In this case the process grows exponentially fast. (2) The eigenvalues of $F$ are on the left half space (i.e., the eigenvalues with negative or zero real parts). The process where all eigenvalues of $F$ have negative real parts is called a stable process and has a unique invariant (i.e., stationary) distribution. In this case the process does not grow. When the eigenvalues of $F$ have zero real parts (i.e., the case of zero eigenvalues and purely imaginary eigenvalues) the process grows polynomially fast. Considering (1) and (2) separately, we first show that an estimator, $\hat{F}$, of $F$ is consistent. We then combine them to present results for the general Ornstein-Uhlenbeck processes. We adopt similar procedure to show the asymptotic efficiency of the estimator.

http://arxiv.org/abs/0805.4535

7107. On the first passage time for Brownian motion subordinated by a Levy process

Author(s): T. R. Hurd and A. Kuznetsov

Abstract: This paper considers the class of L\'evy processes that can be written as a Brownian motion time changed by an independent L\'evy subordinator. Examples in this class include the variance gamma model, the normal inverse Gaussian model, and other processes popular in financial modeling. The question addressed is the precise relation between the standard first passage time and an alternative notion, which we call first passage of the second kind, as suggested by Hurd (2007) and others. We are able to prove that standard first passage time is the almost sure limit of iterations of first passage of the second kind. Many different problems arising in financial mathematics are posed as first passage problems, and motivated by this fact, we are lead to consider the implications of the approximation scheme for fast numerical methods for computing first passage. We find that the generic form of the iteration can be competitive with other numerical techniques. In the particular case of the VG model, the scheme can be further refined to give very fast algorithms.

http://arxiv.org/abs/0805.4618

7108. Numerical Computations for Backward Doubly SDEs and SPDEs

Author(s): Yufeng Shi and Weiqiang Yang and Jing Yuan

Abstract: In this paper we present two numerical schemes of approximating solutions of backward doubly stochastic differential equations (BDSDEs for short). We give a method to discretize a BDSDE. And we also give the proof of the convergence of these two kinds of solutions for BDSDEs respectively. We give a sample of computation of BDSDEs.

http://arxiv.org/abs/0805.4662

7109. Backward SDEs with constrained jumps and Quasi-Variational Inequalities

Author(s): Idris Kharroubi (PMA and CREST) and Jin Ma and Huyen Pham (PMA and CREST) and Jianfeng Zhang

Abstract: We consider a class of backward stochastic differential equations (BSDEs) driven by Brownian motion and Poisson random measure, and subject to constraints on the jump component. We prove the existence and uniqueness of the minimal solution for the BSDEs by using a penalization approach. Moreover, we show that under mild conditions the minimal solutions to these constrained BSDEs can be characterized as the unique viscosity solution of quasi-variational inequalities (QVIs), which leads to a probabilistic representation for solutions to QVIs. Such a representation in particular gives a new stochastic formula for value functions of a class of impulse control problems. As a direct consequence we obtain a numerical scheme for the solution of such QVIs via the simulation of the penalized BSDEs.

http://arxiv.org/abs/0805.4676

7110. Les deux quadrangulations infinies uniformes ont m\^eme loi

Author(s): Laurent M\'enard

Abstract: We prove that the uniform infinite random quadrangulations introduced respectively by Chassaing-Durhuus and Krikun have the same distribution.

http://arxiv.org/abs/0805.4687

7111. Denseness of certain smooth L\'evy functionals in $\DD_{1,2}$

Author(s): Christel Geiss and Eija Laukkarinen

Abstract: The Malliavin derivative for a L\'evy process $(X_t)$ can be defined on the space $\DD_{1,2}$ using a chaos expansion or in the case of a pure jump process also via an increment quotient operator \cite{sole-utzet-vives}. In this paper we define the Malliavin derivative operator $\D$ on the class $\mathcal{S}$ of smooth random variables $f(X_{t_1}, ..., X_{t_n}),$ where $f$ is a smooth function with compact support. We show that the closure of $L_2(\Om) \supseteq \mathcal{S} \stackrel{\D}{\to} L_2(\m\otimes \mass)$ yields to the space $\DD_{1,2}.$ As an application we conclude that Lipschitz functions map from $\DD_{1,2}$ into $\DD_{1,2}.$

http://arxiv.org/abs/0805.4704

7112. Markov chain-based stability analysis of growing networks

Author(s): Zhenting Hou and Jinying Tong and Dinghua Shi

Abstract: From the perspective of probability, the stability of growing network is studied in the present paper. Using the DMS model as an example, we establish a relation between the growing network and Markov process. Based on the concept and technique of first-passage probability in Markov theory, we provide a rigorous proof for existence of the steady-state degree distribution, mathematically re-deriving the exact formula of the distribution. The approach based on Markov chain theory is universal and performs well in a large class of growing networks.

http://arxiv.org/abs/0805.4765

7113. Differentiability of stochastic flow of reflected Brownian motions

Author(s): Krzysztof Burdzy

Abstract: We prove that a stochastic flow of reflected Brownian motions in a smooth multidimensional domain is differentiable with respect to its initial position. The derivative is a linear map represented by a multiplicative functional for reflected Brownian motion. The method of proof is based on excursion theory and analysis of the deterministic Skorokhod equation.

http://arxiv.org/abs/0806.0119

7114. Marking (1,2) Points of the Brownian Web and Applications

Author(s): C. M. Newman (1) and K. Ravishankar (2) and E. Schertzer (1) ((1) Courant Inst. of Mathematical Sciences, NYU, (2) Dept. of Mathematics, SUNY College at New Paltz)

Abstract: The Brownian web (BW), which developed from the work of Arratia and then T\'{o}th and Werner, is a random collection of paths (with specified starting points) in one plus one dimensional space-time that arises as the scaling limit of the discrete web (DW) of coalescing simple random walks. Two recently introduced extensions of the BW, the Brownian net (BN) constructed by Sun and Swart, and the dynamical Brownian web (DyBW) proposed by Howitt and Warren, are (or should be) scaling limits of corresponding discrete extensions of the DW -- the discrete net (DN) and the dynamical discrete web (DyDW). These discrete extensions have a natural geometric structure in which the underlying Bernoulli left or right "arrow" structure of the DW is extended by means of branching (i.e., allowing left and right simultaneously) to construct the DN or by means of switching (i.e., from left to right and vice-versa) to construct the DyDW. In this paper we show that there is a similar structure in the continuum where arrow direction is replaced by the left or right parity of the (1,2) space-time points of the BW (points with one incoming path from the past and two outgoing paths to the future, only one of which is a continuation of the incoming path). We then provide a complete construction of the DyBW and an alternate construction of the BN to that of Sun and Swart by proving that the switching or branching can be implemented by a Poissonian marking of the (1,2) points.

http://arxiv.org/abs/0806.0158

7115. From Black-Scholes and Dupire formulae to last passage times of local martingales. Part A : The infinite time horizon

Author(s): Amel Bentata (PMA) and Marc Yor (PMA and Iuf)

Abstract: These notes are the first half of the contents of the course given by the second author at the Bachelier Seminar (February 8-15-22 2008) at IHP. They also correspond to topics studied by the first author for her Ph.D.thesis.

http://arxiv.org/abs/0806.0239

7116. Backward Stochastic PDEs related to the utility maximization problem

Author(s): M. Mania and R. Tevzadze

Abstract: We study utility maximization problem for general utility functions using dynamic programming approach. We consider an incomplete financial market model, where the dynamics of asset prices are described by an $R^d$-valued continuous semimartingale. Under some regularity assumptions we derive backward stochastic partial differential equation (BSPDE) related directly to the primal problem and show that the strategy is optimal if and only if the corresponding wealth process satisfies a certain forward-SDE. As examples the cases of power, exponential and logarithmic utilities are considered.

http://arxiv.org/abs/0806.0240

7117. Susceptibility in subcritical random graphs

Author(s): Svante Janson and Malwina J. Luczak

Abstract: We study the evolution of the susceptibility in the subcritical random graph $G(n,p)$ as $n$ tends to infinity. We obtain precise asymptotics of its expectation and variance, and show it obeys a law of large numbers. We also prove that the scaled fluctuations of the susceptibility around its deterministic limit converge to a Gaussian law. We further extend our results to higher moments of the component size of a random vertex, and prove that they are jointly asymptotically normal.

http://arxiv.org/abs/0806.0252

7118. Perturbative Approach on Financial Markets

Author(s): Simone Scotti

Abstract: We study the point of transition between complete and incomplete financial models thanks to Dirichlet Forms methods. We apply recent techniques, developped by Bouleau, to hedging procedures in order to perturbate parameters and stochastic processes, in the case of a volatility parameter fixed but uncertain for traders; we call this model Perturbed Black Scholes (PBS) Model. We show that this model can reproduce at the same time a smile effect and a bid-ask spread; we exhibit the volatility function associated to the local-volatility model equivalent to PBS model when vanilla options are concerned. Lastly, we present a connection between Error Theory using Dirichlet Forms and Utility Function Theory.

http://arxiv.org/abs/0806.0287

7119. Risk Premium Impact in the Perturbative Black Scholes Model

Author(s): Luca Regis and Simone Scotti

Abstract: We study the risk premium impact in the Perturbative Black Scholes model. The Perturbative Black Scholes model, developed by Scotti, is a subjective volatility model based on the classical Black Scholes one, where the volatility used by the trader is an estimation of the market one and contains measurement errors. In this article we analyze the correction to the pricing formulas due to the presence of an underlying drift different from the risk free return. We prove that, under some hypothesis on the parameters, if the asset price is a sub-martingale under historical probability, then the implied volatility presents a skewed structure, and the position of the minimum depends on the risk premium $\lambda$.

http://arxiv.org/abs/0806.0307

7120. The growth exponent for planar loop-erased random walk

Author(s): Robert Masson

Abstract: We give a new proof of a result of Kenyon that the growth exponent for loop-erased random walks in two dimensions is 5/4. The proof uses the convergence of LERW to Schramm-Loewner evolution with parameter 2, and is valid for irreducible bounded symmetric random walks on any two-dimensional discrete lattice.

http://arxiv.org/abs/0806.0357

7121. Density fluctuations for a zero-range process on the percolation cluster

Author(s): Patricia Goncalves and Milton Jara

Abstract: We prove that the density fluctuations for a zero-range process evolving on the supercritical percolation cluster are given by a generalized Ornstein-Uhlenbeck process in the space of distributions $\mc S'(\bb R^d)$.

http://arxiv.org/abs/0806.0362

7122. Coarse graining, fractional moments and the critical slope of random copolymers

Author(s): F. Toninelli (Laboratoire de Physique and ENS Lyon and CNRS)

Abstract: For a much-studied model of random copolymer at a selective interface we prove that the slope of the critical curve in the weak-disorder limit is strictly smaller than 1, which is the value given by the annea