Probability Abstracts 100

This document contains abstracts 5997-6227 from September-1-2007 to October-31-2007.
They have been mailed on November 8th, 2007.

5997. A note on the circular law for non-central random matrices

Author(s): Djalil Chafai (IMT and Upte)

Abstract: Let $(X_{i,j})_{1\leq i,j<\infty}$ be an infinite array of i.i.d. complex random variables, with mean $m=0$, variance $\si^2=1$, and say with finite fourth moment. The famous circular law theorem states that the empirical spectral distribution $\frac{1}{n}(\de_{\la_1(\bX)}+...+\de_{\la_n(\bX)})$ of $\bX=(n^{-1/2}X_{i,j})_{1\leq i,j\leq n}$ converges almost surely, as $n\to\infty$, to the uniform law over the unit disc $\{z\in\dC;\ABS{z}\leq 1\}$. For now, most efforts where focused on the improvement of moments hypotheses for the centered case $m=0$. Regarding the non-central case $m\neq0$, Silverstein has already observed that almost surely, the eigenvalue of $\bX$ of largest module goes to $+\infty$ as $n\to\infty$, while the rest of the spectrum remains bounded. We show in this note that the circular law theorem remains valid when $m\neq0$, by using logarithmic potentials and bounds on extremal singular values.

http://arxiv.org/abs/0709.0036

5998. On the precision of the spectral profile

Author(s): Gady Kozma

Abstract: We examine the spectral profile bound of Goel, Montenegro and Tetali for the uniform mixing time of continuous-time random walk in reversible settings. We find that it is precise up to a log log factor, and that this log log factor cannot be improved.

http://arxiv.org/abs/0709.0112

5999. On the shape stability for a growth model

Author(s): M.V. Menshikov and V. Sisko and M. Vachkovskaia

Abstract: We consider a growth model with $n$ nodes. Let us fix ${\cal K}$ subsets $S_i\subset \{1, ..., n\}$, the customers arrive to set $S_i$ at rate $\lambda_i$. A customer arriving to $S_i$ is directed to some node in $S_i$ accordingly to previously chosen rule (policy). Under some natural conditions on $\lambda_i$'s we have found such rules (one of them is Join the Shortest Queue routing policy) for which our model is positive recurrent in shape.

http://arxiv.org/abs/0709.0121

6000. Valuations and dynamic convex risk measures

Author(s): A. Jobert and L. C. G. Rogers

Abstract: This paper approaches the definition and properties of dynamic convex risk measures through the notion of a family of concave valuation operators satisfying certain simple and credible axioms. Exploring these in the simplest context of a finite time set and finite sample space, we find natural risk-transfer and time-consistency properties for a firm seeking to spread its risk across a group of subsidiaries.

http://arxiv.org/abs/0709.0232

6001. Strong Law of Large Numbers for branching diffusions

Author(s): Janos Englander and Simon C. Harris and Andreas E. Kyprianou

Abstract: Let $X$ be the branching particle diffusion corresponding to the operator $Lu+\beta (u^{2}-u)$ on $D\subseteq \mathbb{R}^{d}$ (where $\beta \geq 0$ and $\beta\not\equiv 0$). Let $\lambda_{c}$ denote the generalized principal eigenvalue for the operator $L+\beta $ on $D$ and assume that it is finite. When $\lambda_{c}>0$ and $L+\beta-\lambda_{c}$ satisfies certain spectral theoretical conditions, we prove that the random measure $\exp \{-\lambda_{c}t\}X_{t}$ converges almost surely in the vague topology as $t$ tends to infinity. This result is motivated by a cluster of articles due to Asmussen and Hering dating from the mid-seventies as well as the more recent work concerning analogous results for superdiffusions of \cite{ET,EW}. We extend significantly the results in \cite{AH76,AH77} and include some key examples of the branching process literature. As far as the proofs are concerned, we appeal to modern techniques concerning martingales and `spine' decompositions or `immortal particle pictures'.

http://arxiv.org/abs/0709.0272

6002. Estimating Random Variables from Random Sparse Observations

Author(s): Andrea Montanari

Abstract: Let X_1,...., X_n be a collection of iid discrete random variables, and Y_1,..., Y_m a set of noisy observations of such variables. Assume each observation Y_a to be a random function of some a random subset of the X_i's, and consider the conditional distribution of X_i given the observations, namely \mu_i(x_i)\equiv\prob\{X_i=x_i|Y\} (a posteriori probability). We establish a general relation between the distribution of \mu_i, and the fixed points of the associated density evolution operator. Such relation holds asymptotically in the large system limit, provided the average number of variables an observation depends on is bounded. We discuss the relevance of our result to a number of applications, ranging from sparse graph codes, to multi-user detection, to group testing.

http://arxiv.org/abs/0709.0145

6003. Hydrodynamic behavior of one dimensional subdiffusive exclusion processes with random conductances

Author(s): A. Faggionato and M. Jara and C. Landim

Abstract: Consider a system of particles performing nearest neighbor random walks on the lattice $\ZZ$ under hard--core interaction. The rate for a jump over a given bond is direction--independent and the inverse of the jump rates are i.i.d. random variables belonging to the domain of attraction of an $\a$--stable law, $0<\a<1$. This exclusion process models conduction in strongly disordered one-dimensional media. We prove that, when varying over the disorder and for a suitable slowly varying function $L$, under the super-diffusive time scaling $N^{1 + 1/\alpha}L(N)$, the density profile evolves as the solution of the random equation $\partial_t \rho = \mf L_W \rho$, where $\mf L_W$ is the generalized second-order differential operator $\frac d{du} \frac d{dW}$ in which $W$ is a double sided $\a$--stable subordinator. This result follows from a quenched hydrodynamic limit in the case that the i.i.d. jump rates are replaced by a suitable array $\{\xi_{N,x} : x\in\bb Z\}$ having same distribution and fulfilling an a.s. invariance principle. We also prove a law of large numbers for a tagged particle.

http://arxiv.org/abs/0709.0306

6004. Random walk local time approximated by a Wiener sheet combined with an independent Brownian motion

Author(s): Endre Cs\'aki and Mikl\'os Cs\"org\H{o} and Ant\'onia F\"oldes and P\'al R\'ev\'esz

Abstract: Let $\xi(k,n)$ be the local time of a simple symmetric random walk on the line. We give a strong approximation of the centered local time process $\xi(k,n)-\xi(0,n)$ in terms of a Wiener sheet and an independent Wiener process, time changed by an independent Brownian local time. Some related results and consequences are also established.

http://arxiv.org/abs/0709.0389

6005. Approximation via regularization of the local time of semimartingales and Brownian motion

Author(s): Blandine Berard Bergery (IECN) and Pierre Vallois (IECN)

Abstract: Through a regularization procedure, few approximation schemes of the local time of a large class of one dimensional processes are given. We mainly consider the local time of continuous semimartingales and reversible diffusions, and the convergence holds in ucp sense. In the case of standard Brownian motion, we have been able to determine a rate of convergence in $L^2$, and a.s. convergence of some of our schemes.

http://arxiv.org/abs/0709.0402

6006. Double Clustering and Graph Navigability

Author(s): Oskar Sandberg

Abstract: Graphs are called navigable if one can find short paths through them using only local knowledge. It has been shown that for a graph to be navigable, its construction needs to meet strict criteria. Since such graphs nevertheless seem to appear in nature, it is of interest to understand why these criteria should be fulfilled. In this paper we present a simple method for constructing graphs based on a model where nodes vertices are ``similar'' in two different ways, and tend to connect to those most similar to them - or cluster - with respect to both. We prove that this leads to navigable networks for several cases, and hypothesize that it also holds in great generality. Enough generality, perhaps, to explain the occurrence of navigable networks in nature.

http://arxiv.org/abs/0709.0511

6007. The Laguerre process and generalized Hartman--Watson law

Author(s): Nizar Demni

Abstract: In this paper, we study complex Wishart processes or the so-called Laguerre processes $(X_t)_{t\geq0}$. We are interested in the behaviour of the eigenvalue process; we derive some useful stochastic differential equations and compute both the infinitesimal generator and the semi-group. We also give absolute-continuity relations between different indices. Finally, we compute the density function of the so-called generalized Hartman--Watson law as well as the law of $T_0:=\inf\{t,\det(X_t)=0\}$ when the size of the matrix is 2.

http://www.arxiv.org

6008. Asymptotic expansion and central limit theorem for quadratic variations of Gaussian processes

Author(s): Arnaud Begyn

Abstract: Cohen, Guyon, Perrin and Pontier have given assumptions under which the second-order quadratic variations of a Gaussian process converge almost surely to a deterministic limit. In this paper we present two new convergence results about these variations: the first is a deterministic asymptotic expansion; the second is a central limit theorem. Next we apply these results to identify two-parameter fractional Brownian motion and anisotropic fractional Brownian motion.

http://arxiv.org/abs/0709.0598

6009. Asymptotic normality for the counting process of weak records and \delta-records in discrete models

Author(s): Ra\'ul Gouet and F. Javier L\'opez and Gerardo Sanz

Abstract: Let $\{X_n,n\ge1\}$ be a sequence of independent and identically distributed random variables, taking non-negative integer values, and call $X_n$ a $\delta$-record if $X_n>\max\{X_1,...,X_{n-1}\}+\delta$, where $\delta$ is an integer constant. We use martingale arguments to show that the counting process of $\delta$-records among the first $n$ observations, suitably centered and scaled, is asymptotically normally distributed for $\delta\ne0$. In particular, taking $\delta=-1$ we obtain a central limit theorem for the number of weak records.

http://arxiv.org/abs/0709.0620

6010. Poisson-type deviation inequalities for curved continuous-time Markov chains

Author(s): Ald\'eric Joulin

Abstract: In this paper, we present new Poisson-type deviation inequalities for continuous-time Markov chains whose Wasserstein curvature or $\Gamma$-curvature is bounded below. Although these two curvatures are equivalent for Brownian motion on Riemannian manifolds, they are not comparable in discrete settings and yield different deviation bounds. In the case of birth--death processes, we provide some conditions on the transition rates of the associated generator for such curvatures to be bounded below and we extend the deviation inequalities established [An\'{e}, C. and Ledoux, M. On logarithmic Sobolev inequalities for continuous time random walks on graphs. Probab. Theory Related Fields 116 (2000) 573--602] for continuous-time random walks, seen as models in null curvature. Some applications of these tail estimates are given for Brownian-driven Ornstein--Uhlenbeck processes and $M/M/1$ queues.

http://arxiv.org/abs/0709.0622

6011. On It\^{o}'s formula for elliptic diffusion processes

Author(s): Xavier Bardina and Carles Rovira

Abstract: Bardina and Jolis [Stochastic process. Appl. 69 (1997) 83--109] prove an extension of It\^{o}'s formula for $F(X_t,t)$, where $F(x,t)$ has a locally square-integrable derivative in $x$ that satisfies a mild continuity condition in $t$ and $X$ is a one-dimensional diffusion process such that the law of $X_t$ has a density satisfying certain properties. This formula was expressed using quadratic covariation. Following the ideas of Eisenbaum [Potential Anal. 13 (2000) 303--328] concerning Brownian motion, we show that one can re-express this formula using integration over space and time with respect to local times in place of quadratic covariation. We also show that when the function $F$ has a locally integrable derivative in $t$, we can avoid the mild continuity condition in $t$ for the derivative of $F$ in $x$.

http://arxiv.org/abs/0709.0627

6012. Sample path properties of the local time of multifractional Brownian motion

Author(s): Brahim Boufoussi and Marco Dozzi and Raby Guerbaz

Abstract: We establish estimates for the local and uniform moduli of continuity of the local time of multifractional Brownian motion, $B^H=(B^{H(t)}(t),t\in\mathbb{R}^+)$. An analogue of Chung's law of the iterated logarithm is studied for $B^H$ and used to obtain the pointwise H\"{o}lder exponent of the local time. A kind of local asymptotic self-similarity is proved to be satisfied by the local time of $B^H$.

http://arxiv.org/abs/0709.0637

6013. Limit theorems for functionals on the facets of stationary random tessellations

Author(s): Lothar Heinrich and Hendrik Schmidt and Volker Schmidt

Abstract: We observe stationary random tessellations $X=\{\Xi_n\}_{n\ge1}$ in $\mathbb{R}^d$ through a convex sampling window $W$ that expands unboundedly and we determine the total $(k-1)$-volume of those $(k-1)$-dimensional manifold processes which are induced on the $k$-facets of $X$ ($1\le k\le d-1$) by their intersections with the $(d-1)$-facets of independent and identically distributed motion-invariant tessellations $X_n$ generated within each cell $\Xi_n$ of $X$. The cases of $X$ being either a Poisson hyperplane tessellation or a random tessellation with weak dependences are treated separately. In both cases, however, we obtain that all of the total volumes measured in $W$ are approximately normally distributed when $W$ is sufficiently large. Structural formulae for mean values and asymptotic variances are derived and explicit numerical values are given for planar Poisson--Voronoi tessellations (PVTs) and Poisson line tessellations (PLTs).

http://arxiv.org/abs/0709.0650

6014. On the ruin time distribution for a Sparre Andersen process with exponential claim sizes

Author(s): K. A. Borovkov and D. C. M. Dickson

Abstract: We derive a closed-form (infinite series) representation for the distribution of the ruin time for the Sparre Andersen model with exponentially distributed claims. This extends a recent result of Dickson et al. (2005) for such processes with Erlang inter-claim times. We illustrate our result in the cases of gamma and mixed exponential inter-claim time distributions.

http://arxiv.org/abs/0709.0764

6015. Self-similar stable processes arising from high-density limits of occupation times of particle systems

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

Abstract: We extend results on time-rescaled occupation time fluctuation limits of the $(d,\alpha, \beta)$-branching particle system $(0<\alpha \leq 2, 0<\beta \leq 1)$ with Poisson initial condition. The earlier results in the homogeneous case (i.e., with Lebesgue initial intensity measure) were obtained for dimensions $d>\alpha / \beta$ only, since the particle system becomes locally extinct if $d\le \alpha / \beta$. In this paper we show that by introducing high density of the initial Poisson configuration, limits are obtained for all dimensions, and they coincide with the previous ones if $d>\alpha/\beta$. We also give high-density limits for the systems with finite intensity measures (without high density no limits exist in this case due to extinction); the results are different and harder to obtain due to the non-invariance of the measure for the particle motion. In both cases, i.e., Lebesgue and finite intensity measures, for low dimensions ($d<\alpha(1+\beta)/\beta$ and $d<\alpha(2+\beta)/(1+\beta)$, respectively) the limits are determined by non-L\'evy self-similar stable processes. For the corresponding high dimensions the limits are qualitatively different: ${\cal S}'(R^d)$-valued L\'evy processes in the Lebesgue case, stable processes constant in time on $(0,\infty)$ in the finite measure case. For high dimensions, the laws of all limit processes are expressed in terms of Riesz potentials. If $\beta=1$, the limits are Gaussian. Limits are also given for particle systems without branching, which yields in particular weighted fractional Brownian motions in low dimensions. The results are obtained in the setup of weak convergence of S'(R^d)$-valued processes.

http://arxiv.org/abs/0709.0773

6016. Weak approximation of a fractional SDE

Author(s): Ivan Nourdin (PMA) and Samy Tindel (IECN)

Abstract: In this note, a diffusion approximation result is shown for stochastic differential equations driven by a fractional Brownian motion B with Hurst parameter H>1/3. We shall use a Gaussian regular approximation of B for sake of clarity, and our method of proof will rely on the algebraic integration theory introduced by Gubinelli.

http://arxiv.org/abs/0709.0805

6017. Random colourings of aperiodic graphs: Ergodic and spectral properties

Author(s): Peter M\"uller and Christoph Richard

Abstract: We study randomly coloured graphs embedded into Euclidean space, whose vertex sets are infinite, uniformly discrete subsets of finite local complexity. We construct the appropriate ergodic dynamical systems, explicitly characterise ergodic measures, and prove an ergodic theorem. For covariant operators of finite range defined on those graphs, we show the existence and self-averaging of the integrated density of states, as well as the non-randomness of the spectrum. Our main result establishes Lifshits tails at the lower spectral edge of the graph Laplacian on bond percolation subgraphs, for sufficiently small probabilities. Among other assumptions, its proof requires exponential decay of the cluster-size distribution for percolation on rather general graphs.

http://arxiv.org/abs/0709.0821

6018. Fastest mixing Markov chain on graphs with symmetries

Author(s): Stephen Boyd and Persi Diaconis and Pablo A. Parrilo and Lin Xiao

Abstract: We show how to exploit symmetries of a graph to efficiently compute the fastest mixing Markov chain on the graph (i.e., find the transition probabilities on the edges to minimize the second-largest eigenvalue modulus of the transition probability matrix). Exploiting symmetry can lead to significant reduction in both the number of variables and the size of matrices in the corresponding semidefinite program, thus enable numerical solution of large-scale instances that are otherwise computationally infeasible. We obtain analytic or semi-analytic results for particular classes of graphs, such as edge-transitive and distance-transitive graphs. We describe two general approaches for symmetry exploitation, based on orbit theory and block-diagonalization, respectively. We also establish the connection between these two approaches.

http://arxiv.org/abs/0709.0955

6019. Fault Tolerance in Cellular Automata at High Fault Rates

Author(s): Mark McCann and Nicholas Pippenger

Abstract: A commonly used model for fault-tolerant computation is that of cellular automata. The essential difficulty of fault-tolerant computation is present in the special case of simply remembering a bit in the presence of faults, and that is the case we treat in this paper. We are concerned with the degree (the number of neighboring cells on which the state transition function depends) needed to achieve fault tolerance when the fault rate is high (nearly 1/2). We consider both the traditional transient fault model (where faults occur independently in time and space) and a recently introduced combined fault model which also includes manufacturing faults (which occur independently in space, but which affect cells for all time). We also consider both a purely probabilistic fault model (in which the states of cells are perturbed at exactly the fault rate) and an adversarial model (in which the occurrence of a fault gives control of the state to an omniscient adversary). We show that there are cellular automata that can tolerate a fault rate $1/2 - \xi$ (with $\xi>0$) with degree $O((1/\xi^2)\log(1/\xi))$, even with adversarial combined faults. The simplest such automata are based on infinite regular trees, but our results also apply to other structures (such as hyperbolic tessellations) that contain infinite regular trees. We also obtain a lower bound of $\Omega(1/\xi^2)$, even with purely probabilistic transient faults only.

http://arxiv.org/abs/0709.0967

6020. Generalized solutions of the Cauchy problem for the Navier-Stokes system and diffusion processes

Author(s): S. Albeverio and Ya. Belopolskaya

Abstract: We reduce the construction of a weak solution of the Cauchy problem for the Navier-Stokes system to the construction of a solution to a stochastic problem. Namely, we construct diffusion processes which allow us to obtain a probabilistic representation of a weak (in distributional sense) solution to the Cauchy problem for the Navier- Stokes system.

http://arxiv.org/abs/0709.1008

6021. Critical scaling of stochastic epidemic models

Author(s): Steven P. Lalley

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

http://arxiv.org/abs/0709.1039

6022. Parameter estimation in diagonalizable bilinear stochastic parabolic equations

Author(s): Igor Cialenco and Sergey V. Lototsky

Abstract: A parameter estimation problem is considered for a stochastic parabolic equation with multiplicative noise under the assumption that the equation can be reduced to an infinite system of uncoupled diffusion processes. From the point of view of classical statistics, this problem turns out to be singular not only for the original infinite-dimensional system but also for most finite-dimensional projections. This singularity can be exploited to improve the rate of convergence of traditional estimators as well as to construct completely new closed-form exact estimator.

http://arxiv.org/abs/0709.1135

6023. Characterizing Heavy-Tailed Distributions Induced by Retransmissions

Author(s): Predrag R. Jelenkovic and Jian Tan

Abstract: Consider a generic data unit of random size L that needs to be transmitted over a channel of unit capacity. The channel availability dynamics is modeled as an i.i.d. sequence {A, A_i},i>0 that is independent of L. During each period of time that the channel becomes available, say A_i, we attempt to transmit the data unit. If L< A_i, the transmission was considered successful; otherwise, we wait for the next available period and attempt to retransmit the data from the beginning. We investigate the asymptotic properties of the number of retransmissions N and the total transmission time T until the data is successfully transmitted. In the context of studying the completion times in systems with failures where jobs restart from the beginning, it was shown that this model results in power law and, in general, heavy-tailed delays. The main objective of this paper is to uncover the detailed structure of this class of heavy-tailed distributions induced by retransmissions. More precisely, we study how the functional dependence between P[L>x] and P[A>x] impacts the distributions of N and T. In the space of this functional dependence, we discover several functional criticality points that separate classes of different functional behavior of the distribution of N. We also discuss the engineering implications of our results on communication networks since retransmission strategy is a fundamental component of the existing network protocols on all communication layers, from the physical to the application one.

http://arxiv.org/abs/0709.1138

6024. Divergence of a stationary random vector field can be always positive

Author(s): Boris Tsirelson

Abstract: The divergence of a stationary random vector field at a given point is usually a centered (that is, zero mean) random variable. Strangely enough, it can be equal to 1 almost surely.

http://arxiv.org/abs/0709.1270

6025. Exponential clogging time for a one dimensional DLA

Author(s): Itai Benjamini and Christopher Hoffman

Abstract: When considering DLA on a cylinder it is natural to ask how many particles it takes to clog the cylinder, e.g. modeling clogging of arteries. In this note we formulate a very simple DLA clogging model and establish an exponential lower bound on the number of particles arriving before clogging appears.

http://arxiv.org/abs/0709.1276

6026. Relative and Discrete Utility Maximising Entropy

Author(s): Grzegorz Hara\'nczyk and Wojciech S{\l}omczy\'nski and Tomasz Zastawniak

Abstract: The notion of utility maximising entropy (u-entropy) of a probability density, which was introduced and studied by Slomczynski and Zastawniak (Ann. Prob 32 (2004) 2261-2285, arXiv:math.PR/0410115 v1), is extended in two directions. First, the relative u-entropy of two probability measures in arbitrary probability spaces is defined. Then, specialising to discrete probability spaces, we also introduce the absolute u-entropy of a probability measure. Both notions are based on the idea, borrowed from mathematical finance, of maximising the expected utility of the terminal wealth of an investor. Moreover, u-entropy is also relevant in thermodynamics, as it can replace the standard Boltzmann-Shannon entropy in the Second Law. If the utility function is logarithmic or isoelastic (a power function), then the well-known notions of the Boltzmann-Shannon and Renyi relative entropy are recovered. We establish the principal properties of relative and discrete u-entropy and discuss the links with several related approaches in the literature.

http://arxiv.org/abs/0709.1281

6027. Corners and Records of the Poisson Process in Quadrant

Author(s): Alexander Gnedin

Abstract: The scale-invariant spacings lemma due to Arratia, Barbour and Tavar{\'e} establishes the distributional identity of a self-similar Poisson process and the set of spacings between the points of this process. In this note we connect this result with properties of a certain set of extreme points of the unit Poisson process in the positive quadrant.

http://arxiv.org/abs/0709.1285

6028. A note on the component structure in random graphs with tunable clustering

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

Abstract: We study the component structure in random intersection graphs with tunable clustering, and show that the average degree works as a threshold for a phase transition for the size of the largest component. That is, if the expected degree is less than one, the size of the largest component is $O_{p}(\log n)$, but if the average degree is greater than one, a single large component of size $O_{p}(n)$ emerges, and there will only be a finite number of small vertices of size less than $O_{p}(\log n)$.

http://arxiv.org/abs/0709.1416

6029. Analysis of stochastic fluid queues driven by local time processes

Author(s): Takis Konstantopoulos and Andreas Kyprianou and Marina Sirvio and Paavo Salminen

Abstract: We consider a stochastic fluid queue served by a constant rate server and driven by a process which is the local time of a certain Markov process. Such a stochastic system can be used as a model in a priority service system, especially when the time scales involved are fast. The input (local time) in our model is always singular with respect to the Lebesgue measure which in many applications is ``close'' to reality. We first discuss how to rigorously construct the (necessarily) unique stationary version of the system under some natural stability conditions. We then consider the distribution of performance steady-state characteristics, namely, the buffer content, the idle period and the busy period. These derivations are much based on the fact that the inverse of the local time of a Markov process is a L\'evy process (a subordinator) hence making the theory of L\'evy processes applicable. Another important ingredient in our approach is the Palm calculus coming from the point process point of view.

http://arxiv.org/abs/0709.1456

6030. On differentiability of the Parisi formula

Author(s): Dmitry Panchenko

Abstract: It was proved by Michel Talagrand ("Parisi measures") that Parisi formula is differentiable with respect to inverse temperature parameter. We obtain a simpler proof of this result by using approximate solutions in the Parisi formula and give one example of application to replica symmetry breaking.

http://arxiv.org/abs/0709.1514

6031. American Options under Proportional Transaction Costs: Pricing, Hedging and Stopping Algorithms for Long and Short Positions

Author(s): Alet Roux and Tomasz Zastawniak

Abstract: American options are studied in a general discrete market in the presence of proportional transaction costs, modelled as bid-ask spreads. Pricing algorithms and constructions of hedging strategies, stopping times and martingale representations are presented for short (seller's) and long (buyer's) positions in an American option with an arbitrary payoff. This general approach extends the special cases considered in the literature concerned primarily with computing the prices of American puts under transaction costs by relaxing any restrictions on the form of the payoff, the magnitude of the transaction costs or the discrete market model itself. The largely unexplored case of pricing, hedging and stopping for the American option buyer under transaction costs is also covered. The pricing algorithms are computationally efficient, growing only polynomially with the number of time steps in a recombinant tree model. The stopping times realising the ask (seller's) and bid (buyer's) option prices can differ from one another. The former is generally a so-called mixed (randomised) stopping time, whereas the latter is always a pure (ordinary) stopping time.

http://arxiv.org/abs/0709.1589

6032. Disordered pinning models and copolymers: beyond annealed bounds

Author(s): Fabio Toninelli (ENS Lyon and CNRS)

Abstract: We consider a general model of a disordered copolymer with adsorption. This includes, as particular cases, a generalization of the copolymer at a selective interface introduced by T. Garel et al., pinning and wetting models in various dimensions, and the Poland-Scheraga model of DNA denaturation. We prove a new variational upper bound for the free energy via an estimation of non-integer moments of the partition function. As an application, we show that for strong disorder the quenched critical point differs from the annealed one, e.g., if the disorder distribution is Gaussian. In particular, for pinning/wetting models with loop exponent 0

http://arxiv.org/abs/0709.1629

6033. On the localized phase of a copolymer in an emulsion: supercritical percolation regime

Author(s): Frank den Hollander and Nicolas P\'etr\'elis

Abstract: In this paper we study a two-dimensional directed self-avoiding walk model of a random copolymer in a random emulsion. The copolymer is a random concatenation of monomers of two types, $A$ and $B$, each occurring with density 1/2. The emulsion is a random mixture of liquids of two types, $A$ and $B$, organised in large square blocks occurring with density $p$ and $1-p$, respectively, where $p \in (0,1)$. The copolymer in the emulsion has an energy that is minus $\alpha$ times the number of $AA$-matches minus $\beta$ times the number of $BB$-matches, where without loss of generality the interaction parameters can be taken from the cone $\{(\alpha,\beta)\in\R^2\colon \alpha\geq |\beta|\}$. To make the model mathematically tractable, we assume that the copolymer is directed and can only enter and exit a pair of neighbouring blocks at diagonally opposite corners. In \cite{dHW06}, it was found that in the supercritical percolation regime $p \geq p_c$, with $p_c$ the critical probability for directed bond percolation on the square lattice, the free energy has a phase transition along a curve in the cone that is independent of $p$. At this critical curve, there is a transition from a phase where the copolymer is fully delocalized into the $A$-blocks to a phase where it is partially localized near the $AB$-interface. In the present paper we prove three theorems that complete the analysis of the phase diagram : (1) the critical curve is strictly increasing; (2) the phase transition is second order; (3) the free energy is infinitely differentiable throughout the partially localized phase.

http://arxiv.org/abs/0709.1659

6034. Existence, uniqueness and approximation of stochastic Schrodinger

Author(s): Clement Pellegrini (ICJ)

Abstract: Recent developments in quantum physics make heavy use of so-called ``quantum trajectories''. Mathematically, this theory gives rise to ``stochastic Schr\"odinger equations'', that is, pertubations of Schrodinger-type equations under the form of stochastic differential equations. But such equations are in general not of the usual type as considered in the litterature. They pose a serious problem in terms of: justifying the existence and uniqueness of a solution, justifying the physical pertinence of the equations. In this article we concentrate on a particular case: the diffusive case, for a two-level system. We prove existence and uniqueness of the associated stochastic Schrodinger equation. We physically justify the equations by proving that they are continuous time limit of a concrete physical procedure for obtainig quantum trajectory.

http://arxiv.org/abs/0709.1703

6035. On the asymptotic of likelihood ratios for self-normalized large deviations

Author(s): Zhiyi Chi

Abstract: Motivated by multiple statistical hypothesis testing, we obtain the limit of likelihood ratio of large deviations for self-normalized random variables, specifically, the ratio of $P(\bar X +d/n \ge x_n V)$ to $P(\bar X \ge x_n V)$, as $n\toi$, where $\bar X$ and $V$ are the sample mean and standard deviation of iid $X_1, ..., X_n$, respectively, $d>0$ is a constant and $x_n \toi$. We show that the limit can have a simple form $e^{d/z_0}$, where $z_0$ is the unique maximizer of $z f(x)$ with $f$ the density of $X_i$. The result is applied to derive the minimum sample size per test in order to control the error rate of multiple testing at a target level, when real signals are different from noise signals only by a small shift.

http://arxiv.org/abs/0709.1506

6036. Random walks on quasisymmetric functions

Author(s): Patricia Hersh and Samuel K. Hsiao

Abstract: Conditions are provided under which an endomorphism on quasisymmetric functions gives rise to a left random walk on the descent algebra which is also a lumping of a left random walk on permutations. Spectral results are also obtained. Several well-studied random walks are now realized this way: Stanley's QS-distribution results from endomorphisms given by evaluation maps, a-shuffles result from the a-th convolution power of the universal character, and the Tchebyshev operator of the second kind introduced recently by Ehrenborg and Readdy yields traditional riffle shuffles. A conjecture of Ehrenborg regarding the spectra for a family of random walks on ab-words is proven. A theorem of Stembridge from the theory of enriched P-partitions is also recovered as a special case.

http://arxiv.org/abs/0709.1477

6037. Mean-field conditions for percolation on finite graphs

Author(s): Asaf Nachmias

Abstract: Let G_n be a sequence of finite transitive graphs with vertex degree d=d(n) and |G_n|=n. Denote by p^t(v,v) the return probability after t steps of the non-backtracking random walk on G_n. We show that if p^t(v,v) has quasi-random properties, then critical bond-percolation on G_n has a scaling window of width n^{-1/3}, as it would on a random graph. A consequence of our theorems is that if G_n is a transitive expander family with girth at least (2/3 + eps) \log_{d-1} n, then the size of the largest component in p-bond-percolation with p={1 +O(n^{-1/3}) \over d-1} is roughly n^{2/3}. In particular, bond-percolation on the celebrated Ramanujan graph constructed by Lubotzky, Phillips and Sarnak has the above scaling window. This provides the first examples of quasi-random graphs behaving like random graphs with respect to critical bond-percolation.

http://arxiv.org/abs/0709.1719

6038. A Haar-like Construction for the Ornstein Uhlenbeck Process

Author(s): Thibaud Taillefumier and Marcelo O. Magnasco

Abstract: The classical Haar construction of Brownian motion uses a binary tree of triangular wedge-shaped functions. This basis has compactness properties which make it especially suited for certain classes of numerical algorithms. We present a similar basis for the Ornstein-Uhlenbeck process, in which the basis elements approach asymptotically the Haar functions as the index increases, and preserve the following properties of the Haar basis: all basis elements have compact support on an open interval with dyadic rational endpoints; these intervals are nested and become smaller for larger indices of the basis element, and for any dyadic rational, only a finite number of basis elements is nonzero at that number. Thus the expansion in our basis, when evaluated at a dyadic rational, terminates in a finite number of steps. We prove the covariance formulae for our expansion and discuss its statistical interpretation and connections to asymptotic scale invariance.

http://arxiv.org/abs/0709.1726

6039. On exit times of Levy-driven Ornstein--Uhlenbeck processes

Author(s): K. Borovkov and A. Novikov

Abstract: We prove two martingale identities which involve exit times of Levy-driven Ornstein--Uhlenbeck processes. Using these identities we find an explicit formula for the Laplace transform of the exit time under the assumption that positive jumps of the Levy process are exponentially distributed.

http://arxiv.org/abs/0709.1746

6040. Asymptotics for the Wiener sausage among Poissonian obstacles

Author(s): Ryoki Fukushima

Abstract: We consider the Wiener sausage among Poissonian obstacles. The obstacle is called hard if Brownian motion entering the obstacle is immediately killed, and is called soft if it is killed at certain rate. It is known that Brownian motion conditioned to survive among obstacles is confined in a ball near its starting point. We show the weak law of large numbers, large deviation principle in special cases and the moment asymptotics for the volume of the corresponding Wiener sausage. One of the consequence of our results is that the trajectory of Brownian motion almost fills the confinement ball.

http://arxiv.org/abs/0709.1751

6041. Chaoticity for multi-class systems and exchangeability within classes

Author(s): Carl Graham (CMAP)

Abstract: Under the natural partial exchangeability assumption for multi-class interacting particle systems, we prove that these converge to an independent system with infinite i.i.d. classes if and only if the empirical measure of each class satisfies a weak law of large numbers. This extension of a classical result for exchangeable systems (related to the de Finetti Theorem) is somewhat surprising, since then convergence of each class to infinite i.i.d. particles implies asymptotic independence of particles of different classes.

http://arxiv.org/abs/0709.1918

6042. Symmetric $\alpha$-stable subordinators and Cauchy problems

Author(s): Erkan Nane

Abstract: We survey the results in Nane (E. Nane, Higher order PDE's and iterated processes, Trans. American Math. Soc. (to appear)) and Baeumer, Meerschaert, and Nane (B. Baeumer, M.M. Meerschaert and E. Nane, Brownian subordinators and fractional Cauchy problems: Submitted (2007)) which deal with PDE connection of some iterated processes, and obtain a new probabilistic proof of the equivalence of the higher order PDE's and fractional in time PDE's.

http://arxiv.org/abs/0709.1919

6043. A Clark-Ocone formula in UMD Banach spaces

Author(s): Jan Maas and Jan van Neerven

Abstract: Let H be a separable real Hilbert space and let F = (F_t)_{t\in [0,T]} be the augmented filtration generated by an H-cylindrical Brownian motion W_H on [0,T]. We prove that if E is a UMD Banach space, 1\leq p<\infty, and f\in D^{1,p}(E) is F_T-measurable, then f = \E f + \int_0^T P_F(Df) dW_H where D is the Malliavin derivative and P_F is the projection onto the F-adapted elements in a suitable Banach space of L^p-stochastically integrable L(H,E)-valued processes.

http://arxiv.org/abs/0709.2021

6044. Nonparametric estimation for L\'evy processes from low-frequency observations

Author(s): Michael H. Neumann and Markus Reiss

Abstract: We suppose that a L\'evy process is observed at discrete time points. A rather general construction of minimum-distance estimators is shown to give consistent estimators of the L\'evy-Khinchine characteristics as the number of observations tends to infinity, keeping the observation distance fixed. For a specific $C^2$-criterion this estimator is rate-optimal. The connection with deconvolution and inverse problems is explained. A key step in the proof is a uniform control on the deviations of the empirical characteristic function on the whole real line.

http://arxiv.org/abs/0709.2007

6045. Uniform limit laws of the logarithm for nonparametric estimators of the regression function in presence of censored data

Author(s): Bertrand Maillot and Vivian Viallon

Abstract: In this paper, we establish uniform-in-bandwidth limit laws of the logarithm for nonparametric Inverse Probability of Censoring Weighted (I.P.C.W.) estimators of the multivariate regression function under random censorship. A similar result is deduced for estimators of the conditional distribution function. The uniform-in-bandwidth consistency for estimators of the conditional density and the conditional hazard rate functions are also derived from our main result. Moreover, the logarithm laws we establish are shown to yield almost sure simultaneous asymptotic confidence bands for the functions we consider. Examples of confidence bands obtained from simulated data are displayed.

http://arxiv.org/abs/0709.2050

6046. Pseudo-Maximization and Self-Normalized Processes

Author(s): Victor H. de la Pe\~na and Michael J. Klass and Tze Leung Lai

Abstract: Self-normalized processes are basic to many probabilistic and statistical studies. They arise naturally in the the study of stochastic integrals, martingale inequalities and limit theorems, likelihood-based methods in hypothesis testing and parameter estimation, and Studentized pivots and bootstrap-$t$ methods for confidence intervals. In contrast to standard normalization, large values of the observations play a lesser role as they appear both in the numerator and its self-normalized denominator, thereby making the process scale invariant and contributing to its robustness. Herein we survey a number of results for self-normalized processes in the case of dependent variables and describe a key method called ``pseudo-maximization'' that has been used to derive these results. In the multivariate case, self-normalization consists of multiplying by the inverse of a positive definite matrix (instead of dividing by a positive random variable as in the scalar case) and is ubiquitous in statistical applications, examples of which are given.

http://arxiv.org/abs/0709.2233

6047. On convergence of generators of equilibrium dynamics of hopping particles to generator of a birth-and-death process in continuum

Author(s): E. Lytvynov and P. T. Polara

Abstract: We deal with two following classes of equilibrium stochastic dynamics of infinite particle systems in continuum: hopping particles (also called Kawasaki dynamics), i.e., a dynamics where each particle randomly hops over the space, and birth-and-death process in continuum (or Glauber dynamics), i.e., a dynamics where there is no motion of particles, but rather particles die, or are born at random. We prove that a wide class of Glauber dynamics can be derived as a scaling limit of Kawasaki dynamics. More precisely, we prove the convergence of respective generators on a set of cylinder functions, in the $L^2$-norm with respect to the invariant measure of the processes. The latter measure is supposed to be a Gibbs measure corresponding to a potential of pair interaction, in the low activity-high temperature regime. Our result generalizes that of [Finkelshtein D.L. et al., to appear in Random Oper. Stochastic Equations], which was proved for a special Glauber (Kawasaki, respectively) dynamics.

http://arxiv.org/abs/0709.2284

6048. Branched Polymers

Author(s): Richard Kenyon and Peter Winkler

Abstract: Building on and from the work of Brydges and Imbrie, we give an elementary calculation of the volume of the space of branched polymers of order $n$ in the plane and in 3-space. Our development reveals some more general identities, and allows exact random sampling. In particular we show that a random 3-dimensional branched polymer of order $n$ has diameter of order $\sqrt{n}$.

http://arxiv.org/abs/0709.2325

6049. Queueing for ergodic arrivals and services

Author(s): L. Gyorfi and G. Morvai

Abstract: In this paper we revisit the results of Loynes (1962) on stability of queues for ergodic arrivals and services, and show examples when the arrivals are bounded and ergodic, the service rate is constant, and under stability the limit distribution has larger than exponential tail.

http://arxiv.org/abs/0709.2330

6050. Adjusted Viterbi training for hidden Markov models

Author(s): J. Lember and A. Koloydenko

Abstract: To estimate the emission parameters in hidden Markov models one commonly uses the EM algorithm or its variation. Our primary motivation, however, is the Philips speech recognition system wherein the EM algorithm is replaced by the Viterbi training algorithm. Viterbi training is faster and computationally less involved than EM, but it is also biased and need not even be consistent. We propose an alternative to the Viterbi training -- adjusted Viterbi training -- that has the same order of computational complexity as Viterbi training but gives more accurate estimators. Elsewhere, we studied the adjusted Viterbi training for a special case of mixtures, supporting the theory by simulations. This paper proves the adjusted Viterbi training to be also possible for more general hidden Markov models.

http://arxiv.org/abs/0709.2317

6051. Mutation-selection balance with recombination: convergence to equilibrium for polynomial selection costs

Author(s): Aubrey Clayton and Steven N. Evans

Abstract: We study a (possibly infinite-dimensional) dynamical system model for mutation and selection in the presence of recombination. Some features of the model, such as existence and uniqueness of solutions and convergence to the dynamical system of an approximating sequence of discrete time models, were presented in earlier work by Evans, Steinsaltz, and Wachter for quite general selection costs. Here we establish that the phenomenon of mutation-selection balance occurs in the special case of ``polynomial'' selection costs under mild conditions. That is, we show that the dynamical system has a unique equilibrium and that it converges to this equilibrium from all initial conditions.

http://arxiv.org/abs/0709.1750

6052. A probabilistic proof of Wallis's formula for pi

Author(s): Steven J. Miller

Abstract: Using mostly elementary results and functions from probability, we prove Wallis's formula for pi: pi/2 = prod_n (2n * 2n) / ((2n-1) * (2n+1)). The proof involves normalization constants and the Gamma function, Standard normal, and the Student t-Distribution.

http://arxiv.org/abs/0709.2181

6053. Randomization in C*-algebras and the stability of quantum filters

Author(s): Ramon van Handel

Abstract: The states \rho_i, i=1,2 in the state space S of a C*-algebra A are absolutely continuous if and only if there exist absolutely continuous probability measures \mu_i on S such that \rho_i is the barycenter of \mu_i. This technique allows one to study the transformation of conditional expectations under an absolutely continuous change of state using the classical Bayes formula, which can be exploited to obtain sufficient conditions for the asymptotic stability of quantum Markov filters. In the case that A is finite dimensional, explicitly computable observability criteria are obtained.

http://arxiv.org/abs/0709.2216

6054. Stochastic Variational Partitioned Runge-Kutta Integrators for Constrained Systems

Author(s): Nawaf Bou-Rabee and Houman Owhadi

Abstract: Stochastic variational integrators for constrained, stochastic mechanical systems are developed in this paper. The main results of the paper are twofold: an equivalence is established between a stochastic Hamilton-Pontryagin (HP) principle in generalized coordinates and constrained coordinates via Lagrange multipliers, and variational partitioned Runge-Kutta (VPRK) integrators are extended to this class of systems. Among these integrators are 1/2 and 3/2-order strongly convergent RATTLE-type integrators. We prove order of accuracy of the methods provided. The paper also reviews the deterministic treatment of VPRK integrators from the HP viewpoint.

http://arxiv.org/abs/0709.2222

6055. On rough isometries of Poisson processes on the line

Author(s): Ron Peled

Abstract: Benjamini and Szegedy asked (independently) whether two independent Poisson point processes on the line with the same intensity are rough isometric (quasi-isometric) a.s.. Szegedy conjectured the answer is positive. We prove that this question is equivalent to the following question, given two independent Bernoulli percolations $A$ and $B$ on the natural numbers with 0 adjoined to each of them, there exist constants and probability $p>0$ such that for any $n$ the first $n$ points of $A$ are rough isometric to an initial segment of $B$ with these constants, with 0 mapping to 0 and with probability at least $p$. We then make some progress towards the conjecture by showing that if the constants of the rough isometry are allowed to grow with $n$ then constants of order $\sqrt{\log n}$ will suffice (this quantitative variant was introduced by Benjamini). It appears that this is the first result to improve upon the trivial construction which has constants of order $\log n$. Furthermore, the rough isometry we construct is (weakly) monotone and we include a discussion of monotone rough isometries, their properties and an interesting lattice structure inherent in them.

http://arxiv.org/abs/0709.2383

6056. A new weak approximation scheme of stochastic differential equations by using the Runge-Kutta method

Author(s): Shigeo Kusuoka and Mariko Ninomiya and Syoiti Ninomiya

Abstract: In this paper, authors successfully construct a new algorithm for the new higher order scheme of weak approximation of SDEs. The algorithm presented here is based on [1][2]. Although this algorithm shares some features with the algorithm presented by [3], algorithms themselves are completely different and the diversity is not trivial. They apply this new algorithm to the problem of pricing Asian options under the Heston stochastic volatility model and obtain encouraging results. [1] Shigeo Kusuoka, ``Approximation of Expectation of Diffusion Process and Mathematical Finance,'' Advanced Studies in Pure Mathematics, Proceedings of Final Taniguchi Symposium, Nara 1998 (T. Sunada, ed.), vol. 31 2001, pp. 147--165. [2] Terry Lyons and Nicolas Victoir, ``Cubature on Wiener Space,'' Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 460 (2004), pp. 169--198. [3] Syoiti Ninomiya, Nicolas Victoir, ``Weak approximation of stochastic differential equations and application to derivative pricing'', arXiv:math/0605361v3

http://arxiv.org/abs/0709.2434

6057. On the approach to equilibrium for a polymer with adsorption and repulsion

Author(s): Pietro Caputo and Fabio Martinelli and Fabio Lucio Toninelli

Abstract: We consider paths of a one-dimensional simple random walk conditioned to come back to the origin after L steps (L an even integer). In the 'pinning model' each path \eta has a weight \lambda^{N(\eta)}, where \lambda>0 and N(\eta) is the number of zeros in \eta. When the paths are constrained to be non-negative, the polymer is said to satisfy a hard-wall constraint. Such models are well known to undergo a localization/delocalization transition as the pinning strength \lambda is varied. In this paper we study a natural 'spin flip' dynamics for these models and derive several estimates on its spectral gap and mixing time. In particular, for the system with the wall we prove that relaxation to equilibrium is always at least as fast as in the free case (\lambda=1, no wall), where the gap and the mixing time are known to scale as L^{-2} and L^2\log L, respectively. This improves considerably over previously known results. For the system without the wall we show that the equilibrium phase transition has a clear dynamical manifestation: for \lambda \geq 1 the relaxation is again at least as fast as the diffusive free case, but in the strictly delocalized phase (\lambda < 1) the gap is shown to be O(L^{-5/2}), up to logarithmic corrections. As an application of our bounds, we prove stretched exponential relaxation of local functions in the localized regime.

http://arxiv.org/abs/0709.2612

6058. On homogeneous pinning models and penalizations

Author(s): Mihai Gradinaru (IRMAR) and Samy Tindel (IECN)

Abstract: In this note, we show how the penalization method, introduced in order to describe some non-trivial changes of the Wiener measure, can be applied to the study of some simple polymer models such as the pinning model. The bulk of the analysis is then focused on the study of a martingale which has to be computed as a Markovian limit.

http://arxiv.org/abs/0709.2656

6059. Shannon-MacMillan theorems for discrete random fields along curves and lower bounds for surface-order large deviations

Author(s): Julia Brettschneider

Abstract: The notion of a surface-order specific entropy h_c(P) of a two-dimensional discrete random field P along a curve c is introduced as the limit of rescaled entropies along lattice approximations of the blowups of c. Existence is shown by proving a corresponding Shannon-MacMillan theorem. We obtain a representation of h_c(P) as a mixture of specific entropies along the tangent lines of c. As an application, the specific entropy along curves is used to refine Foellmer and Ort's lower bound for the large deviations of the empirical field of an attractive Gibbs measure from its ergodic behavior in the phase-transition regime.

http://arxiv.org/abs/0709.2662

6060. Regularity of the density for the stochastic heat equation

Author(s): Carl Mueller and David Nualart

Abstract: We study the smoothness of the density of a semilinear heat equation with multiplicative spacetime white noise. Using Malliavin calculus, we reduce the problem to a question of negative moments of solutions of a linear heat equation with multiplicative white noise. Then we settle this question by proving that solutions to the linear equation have negative moments of all orders.

http://arxiv.org/abs/0709.2663

6061. Interacting Brownian motions and the Gross-Pitaevskii formula

Author(s): Stefan Adams and Wolfgang K\"onig

Abstract: We review probabilistic approaches to the Gross-Pitaevskii theory describing interacting dilute systems of particles. The main achievement are large deviations principles for the mean occupation measure of a large system of interacting Brownian motions in a trapping potential. The corresponding rate functions are given as variational problems whose solution provide effective descriptions of the infinite system.

http://arxiv.org/abs/0709.2771

6062. How often does the ratchet click? Facts, heuristics, asymptotics

Author(s): A. Etheridge and P. Pfaffelhuber and A. Wakolbinger

Abstract: The evolutionary force of recombination is lacking in asexually reproducing populations. As a consequence, the population can suffer an irreversible accumulation of deleterious mutations, a phenomenon known as Muller's ratchet. We formulate discrete and continuous time versions of Muller's ratchet. Inspired by Haigh's (1978) analysis of a dynamical system which arises in the limit of large populations, we identify the parameter gamma = N*lambda/(Ns*log(N*lambda)) as most important for the speed of accumulation of deleterious mutations. Here N is population size, s is the selection coefficient and lambda is the deleterious mutation rate. For large parts of the parameter range, measuring time in units of size N, deleterious mutations accumulate according to a power law in N*lambda with exponent gamma if gamma>0.5. For gamma<0.5 mutations cannot accumulate. We obtain diffusion approximations for three different parameter regimes, depending on the speed of the ratchet. Our approximations shed new light on analyses of Stephan et al. (1993) and Gordo & Charlesworth (2000). The heuristics leading to the approximations are supported by simulations.

http://arxiv.org/abs/0709.2775

6063. On the Structure of Quasi-Stationary Competing Particles Systems

Author(s): Louis-Pierre Arguin and Michael Aizenman

Abstract: We study point processes on the real line whose configurations X are locally finite, have a maximum, and evolve through increments which are functions of correlated gaussian variables. The correlations are intrinsic to the points and quantified by a matrix Q={q_ij}. A probability measure on the pair (X,Q) is said to be quasi-stationary if the joint law of the gaps of X and of Q is invariant under the evolution. A known class of universally quasi-stationary processes is given by the Ruelle Probability Cascades (RPC), which are based on hierarchally nested Poisson-Dirichlet processes. It was conjectured that up to some natural superpositions these processes exhausted the class of laws which are robustly quasi-stationary. The main result of this work is a proof of this conjecture for the case where q_ij assume only a finite number of values. The result is of relevance for mean-field spin glass models, where the evolution corresponds to the cavity dynamics, and where the hierarchal organization of the Gibbs measure was first proposed as an ansatz.

http://arxiv.org/abs/0709.2901

6064. On the notion of convex-compactness and its applications

Author(s): Gordan Zitkovic

Abstract: The concept of convex-compactness, weaker than the classical notion of compactness, is introduced and discussed. It is shown that a large class of convex subsets of topological vector spaces shares this property and that is can be used in lieu of compactness in a variety of cases. In particular, we show that bounded-in-probability, convex and closed subsets of the space $\lzer_+(\Omega,\FF,\PP)$ of finite-valued non-negative random variables on a probability space are convex-compact. Applications in optimization and mathematical economics - versions of the Minimax theorem, the fixed-point theorem of Knaster, Kuratowski and Mazurkiewicz as well as the excess-demand theorem of mathematical economics - are provided.

http://arxiv.org/abs/0709.2730

6065. Behavioral Portfolio Selection in Continuous Time

Author(s): Hanqing Jin and Xunyu Zhou

Abstract: This paper formulates and studies a general continuous-time behavioral portfolio selection model under Kahneman and Tversky's (cumulative) prospect theory, featuring S-shaped utility (value) functions and probability distortions. Unlike the conventional expected utility maximization model, such a behavioral model could be easily mis-formulated (a.k.a. ill-posed) if its different components do not coordinate well with each other. Certain classes of an ill-posed model are identified. A systematic approach, which is fundamentally different from the ones employed for the utility model, is developed to solve a well-posed model, assuming a complete market and general It\^o processes for asset prices. The optimal terminal wealth positions, derived in fairly explicit forms, possess surprisingly simple structure reminiscent of a gambling policy betting on a good state of the world while accepting a fixed, known loss in case of a bad one. An example with a two-piece CRRA utility is presented to illustrate the general results obtained, and is solved completely for all admissible parameters. The effect of the behavioral criterion on the risky allocations is finally discussed.

http://arxiv.org/abs/0709.2830

6066. Shape and local growth for multidimensional branching random walks in random environment

Author(s): Francis Comets and Serguei Popov

Abstract: We study branching random walks in random environment on the $d$-dimensional square lattice, $d \geq 1$. In this model, the environment has finite range dependence, and the population size cannot decrease. We prove limit theorems (laws of large numbers) for the set of lattice sites which are visited up to a large time as well as for the local size of the population. The limiting shape of this set is compact and convex, though the local size is given by a concave growth exponent. Also, we obtain the law of large numbers for the logarithm of the total number of particles in the process.

http://arxiv.org/abs/0709.2926

6067. A Unified Approach to Stochastic Evolution Equations Using the Skorokhod Integral

Author(s): S. V. Lototsky and B. L. Rozovskii

Abstract: We study stochastic evolution equations driven by Gaussian noise. The key features of the model are that the operators in the deterministic and stochastic parts can have the same order and the noise can be time-only, space-only, or space-time. Even the simplest equations of this kind do not have a square-integrable solution and must be solved in special weighted spaces. We demonstrate that the Cameron-Martin version of the Wiener chaos decomposition leads to natural weights and a natural replacement of the square integrability condition.

http://arxiv.org/abs/0709.2975

6068. On some probabilistic properties of periodic GARCH processes

Author(s): Abdelouahab Bibi and Abdelhakim Aknouche

Abstract: This paper examines some probabilistic properties of the class of periodic GARCH processes (PGARCH) which feature periodicity in conditional heteroskedasticity. In these models, the parameters are allowed to switch between different regimes, so that their structure shares many properties with periodic ARMA process (PARMA). We examine the strict and second order periodic stationarities, the existence of higher-order moments, the covariance structure, the geometric ergodicity and -mixing of the PGARCH(p,q) process under general and tractable assumptions. Some examples are proposed to illustrate the various concepts.

http://arxiv.org/abs/0709.2983

6069. Some extensions of the uncertainty principle

Author(s): Steeve Zozor and Mariela Portesi and Christophe Vignat

Abstract: In this paper, we study extensions of entropic inequalities recently derived by Bialynicki-Birula [1] and Zozor et al. [2]. These inequalities can be considered as generalizations of the Heisenberg uncertainty principle, since they measure the mutual uncertainty of a wavefunction and its Fourier transform through their associated Renyi entropies with conjugated indexes. We consider here the case where the entropic indexes are not conjugated, in both cases where the state space is discrete and continuous: we discuss the existence of an uncertainty inequality depending on the location of the entropic indexes $\alpha$ and $\beta$ in the plane $(\alpha, \beta)$. The obtained results explain and extend a recent study by Luis [3].

http://arxiv.org/abs/0709.3011

6070. Construction of a stationary queue with impatient customers

Author(s): Pascal Moyal

Abstract: In this paper, we study the stability of queues with impatient customers. Under general stationary ergodic assumptions, we first provide some conditions for such a queue to be regenerative (i.e. to empty a.s. an infinite number of times). In the particular case of a single server operating in First in, First out, we prove the existence (in some cases, on an enlarged probability space) of a stationary workload. This is done by studying a non-monotonic stochastic recursion under the Palm settings, and by stochastic comparison of stochastic recursions.

http://arxiv.org/abs/0709.3012

6071. Random even graphs and the Ising model

Author(s): Geoffrey Grimmett and Svante Janson

Abstract: We explore the relationship between the Ising model with inverse temperature $\beta$, the $q=2$ random-cluster model with edge-parameter $p=1-e^{-2\beta}$, and the random even subgraph with edge-parameter $\frac 12p$. For a planar graph $G$, the boundary edges of the + clusters of the Ising model on the planar dual of $G$ forms a random even subgraph of $G$. A coupling of the random even subgraph of $G$ and the $q=2$ random-cluster model on $G$ is presented, thus extending the above observation to general graphs. A random even subgraph of a planar lattice undergoes a phase transition at the parameter-value $\frac 12 \pc$, where $\pc$ is the critical point of the $q=2$ random-cluster model on the dual lattice. These results are motivated in part by an exploration of the so-called random-current method utilised by Aizenman, Barsky, Fern\'andez and others to solve the Ising model on the $d$-dimensional hypercubic lattice.

http://arxiv.org/abs/0709.3039

6072. The martingale problem for a class of stable-like processes

Author(s): Richard F. Bass and Huili Tang

Abstract: Let $\alpha\in (0,2)$ and consider the operator $$L f(x) =\int [f(x+h)-f(x)-1_{(|h|\leq 1)} \nabla f(x)\cdot h] \frac{A(x,h)}{|h|^{d+\alpha}} dh, $$ where the $\nabla f(x)\cdot h$ term is omitted if $\alpha<1$. We consider the martingale problem corresponding to the operator $L$ and under mild conditions on the function $A$ prove that there exists a unique solution.

http://arxiv.org/abs/0709.3082

6073. Least squares volatility change point estimation for partially observed diffusion processes

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

Abstract: A one dimensional diffusion process $X=\{X_t, 0\leq t \leq T\}$, with drift $b(x)$ and diffusion coefficient $\sigma(\theta, x)=\sqrt{\theta} \sigma(x)$ known up to $\theta>0$, is supposed to switch volatility regime at some point $t^*\in (0,T)$. On the basis of discrete time observations from $X$, the problem is the one of estimating the instant of change in the volatility structure $t^*$ as well as the two values of $\theta$, say $\theta_1$ and $\theta_2$, before and after the change point. It is assumed that the sampling occurs at regularly spaced times intervals of length $\Delta_n$ with $n\Delta_n=T$. To work out our statistical problem we use a least squares approach. Consistency, rates of convergence and distributional results of the estimators are presented under an high frequency scheme. We also study the case of a diffusion process with unknown drift and unknown volatility but constant.

http://arxiv.org/abs/0709.2967

6074. Quasi-maximum likelihood estimation of periodic GARCH processes

Author(s): Abdehakim Aknouche and Abdelouhab Bibi

Abstract: This paper establishes the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) for a GARCH process with periodically time-varying parameters. We first give a necessary and sufficient condition for the existence of a strictly periodically stationary solution for the periodic GARCH (P-GARCH) equation. As a result, it is shown that the moment of some positive order of the P-GARCH solution is finite, under which we prove the strong consistency and asymptotic normality (CAN) of the QMLE without any condition on the moments of the underlying process.

http://arxiv.org/abs/0709.2982

6075. A tail inequality for suprema of unbounded empirical processes with applications to Markov chains

Author(s): Rados{\l}aw Adamczak

Abstract: We present an easy extension of Talagrand's inequality for suprema of empirical processes to processes generated by variables with finite $\psi_1$ norm and apply it to some geometrically ergodic Markov chains to derive versions of Bernstein's and Talagrand's inequalities. We also obtain a bounded difference inequality for symmetric statistics of such Markov chains.

http://arxiv.org/abs/0709.3110

6076. Level sets of the stochastic wave equation driven by a symmetric L\'evy noise

Author(s): Davar Khoshnevisan and Eulalia Nualart

Abstract: We consider the solution $\{u(t,x), t \geq 0, x \in \mathbb{R}\}$ of a system of $d$ linear stochastic wave equations driven by a $d$ dimensional symmetric space-time L\'evy noise. We provide a necessary and sufficient condition, on the characteristic exponent of the L\'evy noise, which describes exactly when the zero set of $u$ is nonvoid. We also compute the Hausdorff dimension of that zero set, when it is nonempty. These results will follow from more general potential-theoretic theorems on level sets of L\'evy sheets.

http://arxiv.org/abs/0709.3165

6077. Maximal regularity for stochastic convolutions driven by Levy noise

Author(s): Zdzislaw Brze\'zniak and Erika Hausenblas

Abstract: We show that the result from Da Prato and Lunardi is valid for stochastic convolutions driven by L\'evy processes.

http://arxiv.org/abs/0709.3179

6078. Distribution functions of linear combinations of lattice polynomials from the uniform distribution

Author(s): Jean-Luc Marichal and Ivan Kojadinovic

Abstract: We give the distribution functions, the expected values, and the moments of linear combinations of lattice polynomials from the uniform distribution. Linear combinations of lattice polynomials, which include weighted sums, linear combinations of order statistics, and lattice polynomials, are actually those continuous functions that reduce to linear functions on each simplex of the standard triangulation of the unit cube. They are mainly used in aggregation theory, combinatorial optimization, and game theory, where they are known as discrete Choquet integrals and Lovasz extensions.

http://arxiv.org/abs/0709.3184

6079. Infinitely divisible distributions over locally compact non-archimedean fields

Author(s): S. V. Ludkovsky

Abstract: The article is devoted to stochastic processes with values in finite-dimensional vector spaces over infinite locally compact fields with non-trivial non-archimedean valuations. Infinitely divisible distributions are investigated. Theorems about their characteristic functionals are proved. Particular cases are demonstrated.

http://arxiv.org/abs/0709.3215

6080. Indeterminacy relations in random dynamics

Author(s): Piotr Garbaczewski

Abstract: We analyze various uncertainty measures for spatial diffusion processes. In this manifestly non-quantum setting, we focus on the existence issue of complementary pairs whose joint dispersion measure has strictly positive lower bound.

http://arxiv.org/abs/cond-mat/0703204

6081. Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time

Author(s): Julien Randon-Furling (LPTMS) and Satya N. Majumdar (LPTMS)

Abstract: We calculate analytically the probability density $P(t_m)$ of the time $t_m$ at which a continuous-time Brownian motion (with and without drift) attains its maximum before passing through the origin for the first time. We also compute the joint probability density $P(M,t_m)$ of the maximum $M$ and $t_m$. In the driftless case, we find that $P(t_m)$ has power-law tails: $P(t_m)\sim t_m^{-3/2}$ for large $t_m$ and $P(t_m)\sim t_m^{-1/2}$ for small $t_m$. In presence of a drift towards the origin, $P(t_m)$ decays exponentially for large $t_m$. The results from numerical simulations are in excellent agreement with our analytical predictions.

http://arxiv.org/abs/0708.2101

6082. Analysis of equilibrium states of Markov solutions to the 3D Navier-Stokes equations driven by additive noise

Author(s): Marco Romito

Abstract: We prove that every Markov solution to the three dimensional Navier-Stokes equation with periodic boundary conditions driven by additive Gaussian noise is uniquely ergodic. The convergence to the (unique) invariant measure is exponentially fast. Moreover, we give a well-posedness criterion for the equations in terms of invariant measures. We also analyse the energy balance and identify the term which ensures equality in the balance.

http://arxiv.org/abs/0709.3267

6083. Line crossing problem for biased monotonic random walks in the plane

Author(s): Mohammad Javaheri

Abstract: In this paper, we study the problem of finding the probability that the two-dimensional (biased) monotonic random walk crosses the line $y=\alpha x+d$, where $\alpha,d \geq 0$. A $\beta$-biased monotonic random walk moves from $(a,b)$ to $(a+1,b)$ or $(a,b+1)$ with probabilities $1/(\beta + 1)$ and $\beta/(\beta + 1)$, respectively. Among our results, we show that if $\beta \geq \lceil \alpha \rceil$, then the $\beta$-biased monotonic random walk, starting from the origin, crosses the line $y=\alpha x+d$ for all $d\geq 0$ with probability 1.

http://arxiv.org/abs/0709.3316

6084. A normal distribution for the disturbance term in regression theory

Author(s): Mr. Lambros Iossif

Abstract: In regression theory, it is stated that the disturbance term follows the normal distribution when the sample size is large. In Professor J.Johnston's words: "In view of the many factors involved, an appeal to the Central Limit Theorem would further suggest a normal distribution for u." This paper includes an elementary proof that the disturbance term follows the normal distribution when n is large.

http://arxiv.org/abs/0709.3414

6085. A note on "signed voter models"

Author(s): E. Andjel and G. Maillard and T.S. Mountford

Abstract: We consider some questions raised by the recent paper of Gantert, L\"owe and Steif (2005) concerning ``signed'' voter models on locally finite graphs. These are voter model like processes with the difference that the edges are considered to be either positive or negative. If an edge between a site $x$ and a site $y$ is negative (respectively positive) the site $y$ will contribute towards the flip rate of $x$ if and only if the two current spin values are equal (respectively opposed).

http://arxiv.org/abs/0709.3468

6086. Sure Wins, Separating Probabilities and the Representation of Linear Functionals

Author(s): Gianluca Cassese

Abstract: We discuss conditions under which a convex cone $\K\subset \R^{\Omega}$ admits a probability $m$ such that $\sup_{k\in \K} m(k)\leq0$. Based on these, we also characterize linear functionals that admit the representation as finitely additive expectations. A version of Riesz decomposition based on this property is obtained as well as a characterisation of positive functionals on the space of integrable functions

http://arxiv.org/abs/0709.3411

6087. Singularity sets of Levy processes

Author(s): Arnaud Durand

Abstract: We completely describe the size and large intersection properties of the Holder singularity sets of Levy processes. We also study the set of times at which a given function cannot be a modulus of continuity of a Levy process. The Holder singularity sets of the sample paths of certain random wavelet series are investigated as well.

http://arxiv.org/abs/0709.3596

6088. Random wavelet series based on a tree-indexed Markov chain

Author(s): Arnaud Durand

Abstract: We study the global and local regularity properties of random wavelet series whose coefficients exhibit correlations given by a tree-indexed Markov chain. We determine the law of the spectrum of singularities of these series, thereby performing their multifractal analysis. We also show that almost every sample path displays an oscillating singularity at almost every point and that the points at which a sample path has at most a given Holder exponent form a set with large intersection.

http://arxiv.org/abs/0709.3597

6089. Random fractals and tree-indexed Markov chains

Author(s): Arnaud Durand

Abstract: We study the size properties of a general model of fractal sets that are based on a tree-indexed family of random compacts and a tree-indexed Markov chain. These fractals may be regarded as a generalization of those resulting from the Moran-like deterministic or random recursive constructions considered by various authors. Among other applications, we consider various extensions of Mandelbrot's fractal percolation process.

http://arxiv.org/abs/0709.3598

6090. Existence, uniqueness and approximation for stochastic Schrodinger

Author(s): Clement Pellegrini (ICJ)

Abstract: In quantum physics, recent investigations deal with the so-called "quantum trajectory" theory. Heuristic rules are usually used to give rise to "stochastic Schrodinger equations" which are stochastic differential equations of non-usual type describing the physical models. These equations pose tedious problems in terms of mathematical justification: notion of solution, existence, uniqueness, justification... In this article, we concentrate on a particular case: the Poisson case. Random measure theory is used in order to give rigorous sense to such equations. We prove existence and uniqueness of a solution for the associated stochastic equation. Furthermore, the stochastic model is physically justified by proving that the solution can be obtained as a limit of a concrete discrete time physical model.

http://arxiv.org/abs/0709.3713

6091. On the path structure of a semimartingale arising from monotone probability theory

Author(s): Alexander C. R. Belton

Abstract: Let X be the unique normal martingale such that X_0 = 0 and d[X]_t = (1 - t - X_{t-}) dX_t + dt and let Y_t := X_t + t for all t >= 0; the semimartingale Y arises in quantum probability, where it is the monotone-independent analogue of the Poisson process. The trajectories of Y are examined and various probabilistic properties are derived; in particular, the level set {t >= 0 : Y_t = 1} is shown to be non-empty, compact, perfect and of zero Lebesgue measure. The local times of Y are found to be trivial except for that at level 1; consequently, the jumps of Y are not locally summable.

http://arxiv.org/abs/0709.3788

6092. On Real-Time Communication Systems with Noisy Feedback

Author(s): Aditya Mahajan and Demosthenis Teneketzis

Abstract: We consider a real-time communication system with noisy feedback consisting of a Markov source, a forward and a backward discrete memoryless channels, and a receiver with finite memory. The objective is to design an optimal communication strategy (that is, encoding, decoding, and memory update strategies) to minimize the total expected distortion over a finite horizon. We present a sequential decomposition for the problem, which results in a set of nested optimality equations to determine optimal communication strategies. This provides a systematic methodology to determine globally optimal joint source-channel encoding and decoding strategies for real-time communication systems with noisy feedback.

http://arxiv.org/abs/0709.3753

6093. Mass transport and variants of the logarithmic Sobolev inequality

Author(s): Franck Barthe and Alexander V. Kolesnikov

Abstract: We develop the optimal transportation approach to modified log-Sobolev inequalities and to isoperimetric inequalities. Various sufficient conditions for such inequalities are given. Some of them are new even in the classical log-Sobolev case. The idea behind many of these conditions is that measures with a non-convex potential may enjoy such functional inequalities provided they have a strong integrability property that balances the lack of convexity. In addition, several known criteria are recovered in a simple unified way by transportation methods and generalized to the Riemannian setting.

http://arxiv.org/abs/0709.3890

6094. Variations and estimators for the selfsimilarity order through Malliavin calculus

Author(s): Ciprian Tudor (CES) and Frederi Viens

Abstract: Using multiple stochastic integrals, we analyze the asymptotic behavior of quadratic variations for Gaussian and non-Gaussian selfsimilar processes. We apply our results to the study of statistical estimators for the selfsimilarity index.

http://arxiv.org/abs/0709.3896

6095. Non-central convergence of multiple integrals

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

Abstract: Fix $\nu >0$, denote by $G(\nu/2)$ a Gamma random variable with parameter $\nu/2$, and let $n\geq2$ be an even integer. Consider a sequence $\{F_k\}_{k\geq 1}$ of square integrable random variables, belonging to the $n$th Wiener chaos of a given Gaussian process and with variance converging to $2\nu$. We prove that $\{F_k\}_{k\geq 1}$ converges in distribution to $2G(\nu/2) - \nu$, if, and only if, $E(F_k^4)-12 E(F_{k}^3)\to 12\nu^2-48\nu$. Observe that, if $\nu\geq 1$ is an integer, then $2G(\nu/2) - \nu$ has a centered $\chi^2$ law with $\nu$ degrees of freedom. Our approach involves the techniques of Malliavin calculus recently developed by Nualart and Ortiz-Latorre (2007). We also obtain some multidimensional non-central limit theorems, as well as several equivalent conditions in terms of Malliavin derivatives and norms of contraction operators. Our results should be compared with the main findings by Nualart and Peccati (2005), where it is shown that a normalized sequence of multiple Wiener-It\^{o} integrals converges in law to a Gaussian random variable if, and only if, the sequence of their fourth moments converges to 3.

http://arxiv.org/abs/0709.3903

6096. The Circular Law for Random Matrices

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

Abstract: We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent real entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries. We assume that the entries have moment of order $\E|X_{jk}|^2\phi(x)$, with some positive function $\phi(x)$ which is growing of order $(\log(1+|x|))^7$, or that they are sparsely non-zero. The results are based on and extend previous work of Bai, Rudelson and the authors.

http://arxiv.org/abs/0709.3995

6097. Geographic Gossip: Efficient Averaging for Sensor Networks

Author(s): Alexandros G. Dimakis and Anand D. Sarwate and Martin J. Wainwright

Abstract: Gossip algorithms for distributed computation are attractive due to their simplicity, distributed nature, and robustness in noisy and uncertain environments. However, using standard gossip algorithms can lead to a significant waste in energy by repeatedly recirculating redundant information. For realistic sensor network model topologies like grids and random geometric graphs, the inefficiency of gossip schemes is related to the slow mixing times of random walks on the communication graph. We propose and analyze an alternative gossiping scheme that exploits geographic information. By utilizing geographic routing combined with a simple resampling method, we demonstrate substantial gains over previously proposed gossip protocols. For regular graphs such as the ring or grid, our algorithm improves standard gossip by factors of $n$ and $\sqrt{n}$ respectively. For the more challenging case of random geometric graphs, our algorithm computes the true average to accuracy $\epsilon$ using $O(\frac{n^{1.5}}{\sqrt{\log n}} \log \epsilon^{-1})$ radio transmissions, which yields a $\sqrt{\frac{n}{\log n}}$ factor improvement over standard gossip algorithms. We illustrate these theoretical results with experimental comparisons between our algorithm and standard methods as applied to various classes of random fields.

http://arxiv.org/abs/0709.3921

6098. Uniqueness of solutions of stochastic differential equations

Author(s): A. M. Davie

Abstract: We consider a d-dimensional stochastic differential equation with additive noise and a drift coefficient which is assumed only to be a bounded Borel function. We show that, for almost all choices of the driving Brownian path, the equation has a unique solution.

http://arxiv.org/abs/0709.4147

6099. Threshold phenomena on product spaces: BKKKL revisited (once more)

Author(s): Raphael Rossignol

Abstract: We revisit the work of Bourgain, Kahn, Kalai, Katznelson and Linial (1992) -- referred to as ``BKKKL'' in the title -- about influences on Boolean functions in order to give a precise statement of threshold phenomenon on the product space $\{1,..., r\}^\NN$, generalizing one of the main results of a paper by Talagrand (1994).

http://arxiv.org/abs/0709.4178

6100. Spherical Model in a Random Field

Author(s): A.E. Patrick

Abstract: We investigate the properties of the Gibbs states and thermodynamic observables of the spherical model in a random field. We show that on the low-temperature critical line the magnetization of the model is not a self-averaging observable, but it self-averages conditionally. We also show that an arbitrarily weak homogeneous boundary field dominates over fluctuations of the random field once the model transits into a ferromagnetic phase. As a result, a homogeneous boundary field restores the conventional self-averaging of thermodynamic observables, like the magnetization and the susceptibility. We also investigate the effective field created at the sites of the lattice by the random field, and show that at the critical temperature of the spherical model the effective field undergoes a transition into a phase with long-range correlations $\sim r^{4-d}$.

http://arxiv.org/abs/0709.3579

6101. Weak convergence of Vervaat and Vervaat Error processes of long-range dependent sequences

Author(s): Mikl\'os Cs\"org\Ho and Rafa{\l} Kulik

Abstract: Following Cs\"{o}rg\H{o}, Szyszkowicz and Wang (Ann. Statist. {\bf 34}, (2006), 1013--1044) we consider a long range dependent linear sequence. We prove weak convergence of the uniform Vervaat and the uniform Vervaat error processes, extending their results to distributions with unbounded support and removing normality assumption.

http://arxiv.org/abs/0709.4285

6102. Modulated Branching Processes, Origins of Power Laws and Queueing Duality

Author(s): Predrag R. Jelenkovic and Jian Tan

Abstract: Power law distributions have been repeatedly observed in a wide variety of socioeconomic, biological and technological areas. In the vast majority of these observations, e.g., city populations and sizes of living organisms, the objects of interest evolve due to the replication of their many independent components, e.g., births-deaths of individuals and replications of cells. Furthermore, the rates of replication of the many components are often controlled by exogenous parameters causing periods of expansion and contraction, e.g., baby booms and busts, economic booms and recessions, etc. In addition, the sizes of these objects often either have reflective lower boundaries, e.g., cities do not fall bellow a certain size, low income individuals are subsidized by the government, companies are protected by bankruptcy laws, etc; or have porous/absorbing lower boundaries, e.g., cities may degenerate, bankruptcy protections may fail and companies can be liquidated. Hence, it is natural to propose reflected modulated branching processes as generic models for many of the preceding observations of power laws that are typically observed in proportional growth environments. Indeed, our main results show that the proposed mathematical models result in power law distributions under quite general polynomial Gartner-Ellis conditions. The generality of our results could explain the ubiquitous nature of power law distributions. Furthermore, an informal interpretation of our main results suggests that alternating periods of expansion and reduction, e.g., economic booms and recessions, are primarily responsible for the appearance of power law distributions.

http://arxiv.org/abs/0709.4297

6103. Noncommutative Brownian motions with Kesten distributions and related Poisson processes

Author(s): Romuald Lenczewski and Rafal Salapata

Abstract: We introduce and study a noncommutative two-parameter family of noncommutative Brownian motions in the free Fock space. They are associated with Kesten laws and give a continuous interpolation between Brownian motions in free probability and monotone probability. The combinatorics of our model is based on ordered non-crossing partitions, in which to each such partition $P$ we assign a weight depending on the numbers of disorders and orders in $P$ related to the natural partial order on the set of blocks of $P$ implemented by the relation of being inner or outer. In particular, we obtain a simple relation between Delaney's numbers (related to inner blocks in non-crossing partitions) and generalized Euler's numbers (related to orders and disorders in ordered non-crossing partitions). An important feature of our interpolation is that the mixed moments of the corresponding creation and annihilation processes also reproduce their monotone and free counterparts, which does not take place in other interpolations. The same combinatorics is used to construct an interpolation between free and monotone Poisson processes.

http://arxiv.org/abs/0709.4334

6104. A Convex Stochastic Optimization Problem Arising from Portfolio Selection

Author(s): Hanqing Jin and Zuo Quan Xu and Xun Yu Zhou

Abstract: A continuous-time financial portfolio selection model with expected utility maximization typically boils down to solving a (static) convex stochastic optimization problem in terms of the terminal wealth, with a budget constraint. In literature the latter is solved by assuming {\it a priori} that the problem is well-posed (i.e., the supremum value is finite) and a Lagrange multiplier exists (and as a consequence the optimal solution is attainable). In this paper it is first shown, via various counter-examples, neither of these two assumptions needs to hold, and an optimal solution does not necessarily exist. These anomalies in turn have important interpretations in and impacts on the portfolio selection modeling and solutions. Relations among the non-existence of the Lagrange multiplier, the ill-posedness of the problem, and the non-attainability of an optimal solution are then investigated. Finally, explicit and easily verifiable conditions are derived which lead to finding the unique optimal solution.

http://arxiv.org/abs/0709.4467

6105. Generalized Descents and Normality

Author(s): Miklos Bona

Abstract: We use Janson's dependency criterion to prove that the distribution of $d$-descents of permutations of length $n$ converge to a normal distribution as $n$ goes to infinity. We show that this remains true even if $d$ is allowed to grow with $n$, up to a certain degree.

http://arxiv.org/abs/0709.4483

6106. The Dirichlet Markov Ensemble

Author(s): Djalil Chafai (IMT and Upte)

Abstract: We equip the polytope of $n\times n$ Markov matrices with the normalized trace of the Lebesgue measure of $R^{n^2}$. This probability space provides random Markov matrices, with i.i.d. rows following the Dirichlet distribution of mean $(1/n,...,1/n)$. We show that if $M$ is such a random matrix, then the empirical spectral distribution of $nMM^\top$ tends as $n\to\infty$ to a Marchenko-Pastur distribution. This phenomenon complements an already known result on the sub-dominant eigenvalue of certain random matrices with independent rows, which suggests that the typical spectral gap of a uniform random Markov matrix is of order $1-1/\sqrt{n}$ when $n$ is large. However, some computer simulations reveal striking asymptotic spectral properties of such random matrices, still waiting for a rigorous mathematical analysis. In particular, we conjecture that the empirical distribution of the complex spectrum of $\sqrt{n}M$ tends as $n\to\infty$ to the uniform distribution on the unit disc of the complex plane.

http://arxiv.org/abs/0709.4678

6107. High-order accurate implicit methods for the pricing of barrier options

Author(s): J.C. Ndogmo and D. B. Ntwiga

Abstract: This paper deals with a high-order accurate implicit finite-difference approach to the pricing of barrier options. In this way various types of barrier options are priced, including barrier options paying rebates, and options on dividend-paying-stocks. Moreover, the barriers may be monitored either continuously or discretely. In addition to the high-order accuracy of the scheme, and the stretching effect of the coordinate transformation, the main feature of this approach lies on a probability-based optimal determination of boundary conditions. This leads to much faster and accurate results when compared with similar pricing approaches. The strength of the present scheme is particularly demonstrated in the valuation of discretely monitored barrier options where it yields values closest to those obtained from the only semi-analytical valuation method available.

http://arxiv.org/abs/0710.0069

6108. Applications of integral transforms in fractional diffusion processes

Author(s): Francesco Mainardi

Abstract: The fundamental solution (Green function) for the Cauchy problem of the space-time fractional diffusion equation is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. Then, by using the Mellin transform, a general representation of the Green function in terms of Mellin-Barnes integrals in the complex plane is derived. This allows us to obtain a suitable computational form of the Green function in the space-time domain and to analyse its probability interpretation.

http://arxiv.org/abs/0710.0145

6109. Selfdecomposability and semi-selfdecomposability in subordination of cone-parameter convolution semigroups

Author(s): Ken-iti Sato

Abstract: Extension of two known facts concerning subordination is made. The first fact is that, in subordination of 1-dimensional Brownian motion with drift, selfdecomposability is inherited from subordinator to subordinated. This is extended to subordination of cone-parameter convolution semigroups. The second fact is that, in subordination of strictly stable cone-parameter convolution semigroups on $\mathbb{R}^d$, selfdecomposability is inherited from subordinator to subordinated. This is extended to semi-selfdecomposability.

http://arxiv.org/abs/0710.0193

6110. Branching diffusions, superdiffusions and random media

Author(s): J\'anos Engl\"ander

Abstract: Spatial branching processes became increasingly popular in the past decades, not only because of their obvious connection to biology, but also because superprocesses are intimately related to nonlinear partial differential equations. Another hot topic in today's research in probability theory is `random media', including the now classical problems on `Brownian motion among obstacles' and the more recent `random walks in random environment' and `catalytic branching' models. These notes aim to give a gentle introduction into some topics in spatial branching processes and superprocesses in deterministic environments (sections 2-6) and in random media (sections 7-11).

http://arxiv.org/abs/0710.0236

6111. Central limits and homogenization in random media

Author(s): Guillaume Bal

Abstract: We consider the perturbation of elliptic operators of the form $-\Delta + q_0$ by random, rapidly varying, sufficiently mixing, potentials of the form $q(\frac{\bx}\eps,\omega)$. We analyze the source and spectral problems associated to such operators and show that the properly renormalized difference between the perturbed and unperturbed solutions may be written asymptotically as $\eps\to0$ as explicit Gaussian processes. Such results may be seen as central limit corrections to the homogenization (law of large numbers) process. Similar results are derived for more general elliptic equations in one dimension of space. The results are based on the availability of a rapidly converging integral formulation for the perturbed solutions and on the use of classical central limit results for random processes with appropriate mixing conditions.

http://arxiv.org/abs/0710.0363

6112. Convergence of simple random walks on random discrete trees to Brownian motion on the continuum random tree

Author(s): David Croydon

Abstract: In this article it is shown that the Brownian motion on the continuum random tree is the scaling limit of the simple random walks on any family of discrete $n$-vertex ordered graph trees whose search-depth functions converge to the Brownian excursion as $n\to\infty$. We prove both a quenched version (for typical realisations of the trees) and an annealed version (averaged over all realisations of the trees) of our main result. The assumptions of the article cover the important example of simple random walks on the trees generated by the Galton-Watson branching process, conditioned on the total population size.

http://arxiv.org/abs/0710.0460

6113. Steady-state analysis of a multi-server queue in the Halfin-Whitt regime

Author(s): David Gamarnik and Petar Momcilovic

Abstract: We examine a multi-server queue in the Halfin-Whitt (Quality- and Efficiency-Driven) regime: as the number of servers $n$ increases, the utilization approaches 1 from below at the rate $\Theta(1/\sqrt{n})$. The arrival process is renewal and service times have a lattice-valued distribution with a finite support. We consider the steady-state distribution of the queue length and waiting time in the limit as the number of servers $n$ increases indefinitely. The queue length distribution, in the limit as $n\to\infty$, is characterized in terms of the stationary distribution of an explicitly constructed Markov chain. As a consequence, the steady-state queue length and waiting time scale as $\Theta(\sqrt{n})$ and $\Theta(1/\sqrt{n})$ as $n\to\infty$, respectively. Moreover, an explicit expression for the critical exponent is derived for the moment generating function of a limiting (scaled) steady-state queue length. This exponent depends on three parameters: the amount of spare capacity and the coefficients of variation of interarrival and service times. Interestingly, it matches an analogous exponent corresponding to a single-server queue in the conventional heavy-traffic regime. The results are derived by analyzing Lyapunov functions.

http://arxiv.org/abs/0710.0654

6114. Differential equations driven by rough paths: an approach via discrete approximation

Author(s): A. M. Davie

Abstract: A theory of differential equations driven by a non-differentiable path has recently been developed by Lyons. We develop an alternative approach to this theory, using (modified Euler approximations), and investigate its applicability to stochastic differential equations driven by Brownian motion. We also give some other examples showing that the main results are reasonably sharp.

http://arxiv.org/abs/0710.0772

6115. Monte Carlo Methods and Path-Generation techniques for Pricing Multi-asset Path-dependent Options

Author(s): Piergiacomo Sabino (Dipartimento di Matematica and Universit\`a degli Studi di Bari)

Abstract: We consider the problem of pricing path-dependent options on a basket of underlying assets using simulations. As an example we develop our studies using Asian options. Asian options are derivative contracts in which the underlying variable is the average price of given assets sampled over a period of time. Due to this structure, Asian options display a lower volatility and are therefore cheaper than their standard European counterparts. This paper is a survey of some recent enhancements to improve efficiency when pricing Asian options by Monte Carlo simulation in the Black-Scholes model. We analyze the dynamics with constant and time-dependent volatilities of the underlying asset returns. We present a comparison between the precision of the standard Monte Carlo method (MC) and the stratified Latin Hypercube Sampling (LHS). In particular, we discuss the use of low-discrepancy sequences, also known as Quasi-Monte Carlo method (QMC), and a randomized version of these sequences, known as Randomized Quasi Monte Carlo (RQMC). The latter has proven to be a useful variance reduction technique for both problems of up to 20 dimensions and for very high dimensions. Moreover, we present and test a new path generation approach based on a Kronecker product approximation (KPA) in the case of time-dependent volatilities. KPA proves to be a fast generation technique and reduces the computational cost of the simulation procedure.

http://arxiv.org/abs/0710.0850

6116. Lectures on two-dimensional critical percolation

Author(s): Wendelin Werner

Abstract: This is the preliminary version of the notes corresponding to the course given at the IAS-Park City graduate summer school in July 2007.

http://arxiv.org/abs/0710.0856

6117. Dynamic Programming Optimization over Random Data: the Scaling Exponent for Near-optimal Solutions

Author(s): David J. Aldous and Charles Bordenave and Marc Lelarge

Abstract: A very simple example of an algorithmic problem solvable by dynamic programming is to maximize, over sets A in {1,2,...,n}, the objective function |A| - \sum_i \xi_i 1(i \in A,i+1 \in A) for given \xi_i > 0. This problem, with random (\xi_i), provides a test example for studying the relationship between optimal and near-optimal solutions of combinatorial optimization problems. We show that, amongst solutions differing from the optimal solution in a small proportion \delta of places, we can find near-optimal solutions whose objective function value differs from the optimum by a factor of order \delta^2 but not smaller order. We conjecture this relationship holds widely in the context of dynamic programming over random data, and Monte Carlo simulations for the Kauffman-Levin NK model are consistent with the conjecture. This work is a technical contribution to a broad program initiated in Aldous-Percus (2003) of relating such scaling exponents to the algorithmic difficulty of optimization problems.

http://arxiv.org/abs/0710.0857

6118. A new technique for proving uniqueness for martingale problems

Author(s): Richard F. Bass and Edwin A. Perkins

Abstract: A new technique for proving uniqueness of martingale problems is introduced. The method is illustrated in the context of elliptic diffusions in $R^d$.

http://arxiv.org/abs/0710.0860

6119. Subadditivity of the entropy and its relation to Brascamp-Lieb type inequalities

Author(s): Eric A. Carlen and Dario Cordero-Erausquin

Abstract: We prove a general duality result showing that a Brascamp--Lieb type inequality is equivalent to an inequality expressing subadditivity of the entropy, with a complete correspondence of best constants and cases of equality. This open a new approach to the proof of Brascamp--Lieb type inequalities, via subadditivity of the entropy. We illustrate the utility of this approach by proving a general inequality expressing the subadditivity property of the entropy on $\R^n$, and fully determining the cases of equality. As a consequence of the duality mentioned above, we obtain a simple new proof of the classical Brascamp--Lieb inequality, and also a fully explicit determination of all of the cases of equality. We also deduce several other consequences of the general subadditivity inequality, including a generalization of Hadamard's inequality for determinants. Finally, we also prove a second duality theorem relating superadditivity of the Fisher information and a sharp convolution type inequality for the fundamental eigenvalues of Schr\"odinger operators. Though we focus mainly on the case of random variables in $\R^n$ in this paper, we discuss extensions to other settings as well.

http://arxiv.org/abs/0710.0870

6120. The Starting and Stopping Problem under Knightian Uncertainty and Related Systems of Reflected BSDEs

Author(s): Said Hamadene and Jianfeng Zhang

Abstract: This article deals with the starting and stopping problem under Knightian uncertainty, i.e., roughly speaking, when the probability under which the future evolves is not exactly known. We show that the lower price of a plant submitted to the decisions of starting and stopping is given by a solution of a system of two reflected backward stochastic differential equations (BSDEs for short). We solve this latter system and we give the expression of the optimal strategy. Further we consider a more general system of $m$ ($m\geq 2$) reflected BSDEs with interconnected obstacles. Once more we show existence and uniqueness of the solution of that system.

http://arxiv.org/abs/0710.0908

6121. Stabilizability and percolation in the infinite volume sandpile model

Author(s): Anne Fey and Ronald Meester and Frank Redig

Abstract: We study the sandpile model in infinite volume on $\Zd$. In particular we are interested in relations between the density $\rho$ of a stationary measure $\mu$ on initial configurations, and the question whether or not initial configurations are stabilizable. We prove that stabilizability does not depend on the particular stabilizability rule we adopt. In $d=1$ and $\mu$ a product measure with $\rho=1$ (the known critical value for stabilizability in $d=1$) with a positive density o