Showing papers in "Electronic Journal of Probability in 2009"
TL;DR: In this article, three different fractional versions of the standard Poisson process and some related results concerning the distribution of order statistics and the compound poisson process are presented, and a planar random motion described by a particle moving at finite velocity and changing direction at times spaced by fractional Poisson processes is presented.
Abstract: We present three different fractional versions of the Poisson process and some related results concerning the distribution of order statistics and the compound Poisson process. The main version is constructed by considering the difference-differential equation governing the distribution of the standard Poisson process, $ N(t),t>0$, and by replacing the time-derivative with the fractional Dzerbayshan-Caputo derivative of order $
u\in(0,1]$. For this process, denoted by $\mathcal{N}_
u(t),t>0,$ we obtain an interesting probabilistic representation in terms of a composition of the standard Poisson process with a random time, of the form $\mathcal{N}_
u(t)= N(\mathcal{T}_{2
u}(t)),$ $t>0$. The time argument $\mathcal{T}_{2
u }(t),t>0$, is itself a random process whose distribution is related to the fractional diffusion equation. We also construct a planar random motion described by a particle moving at finite velocity and changing direction at times spaced by the fractional Poisson process $\mathcal{N}_
u.$ For this model we obtain the distributions of the random vector representing the position at time $t$, under the condition of a fixed number of events and in the unconditional case. For some specific values of $
u\in(0,1]$ we show that the random position has a Brownian behavior (for $
u =1/2$) or a cylindrical-wave structure (for $
u =1$).
233 citations
TL;DR: In this article, the authors considered nonlinear parabolic SPDEs of the form ǫ √ √ t u = √ T u + \sigma(u)dot w, where À t u denotes space-time white noise, and è t u is the generator of a L'evy process.
Abstract: We consider nonlinear parabolic SPDEs of the form $\partial_t u={\cal L} u + \sigma(u)\dot w$, where $\dot w$ denotes space-time white noise, $\sigma:R\to R$ is [globally] Lipschitz continuous, and $\cal L$ is the $L^2$-generator of a L'evy process. We present precise criteria for existence as well as uniqueness of solutions. More significantly, we prove that these solutions grow in time with at most a precise exponential rate. We establish also that when $\sigma$ is globally Lipschitz and asymptotically sublinear, the solution to the nonlinear heat equation is ``weakly intermittent,'' provided that the symmetrization of $\cal L$ is recurrent and the initial data is sufficiently large. Among other things, our results lead to general formulas for the upper second-moment Liapounov exponent of the parabolic Anderson model for $\cal L$ in dimension $(1+1)$. When ${\cal L}=\kappa\partial_{xx}$ for $\kappa>0$, these formulas agree with the earlier results of statistical physics (Kardar (1987), Krug and Spohn (1991), Lieb and Liniger (1963)), and also probability theory (Bertini and Cancrini (1995), Carmona and Molchanov (1994)) in the two exactly-solvable cases. That is when $u_0=\delta_0$ or $u_0\equiv 1$; in those cases the moments of the solution to the SPDE can be computed (Bertini and Cancrini (1995)).
152 citations
TL;DR: In this article, a simple trick of exploding vertices instead of deleting them, enables us to derive results from known results for random graphs with given vertex degrees, which is used to study existence of giant component and existence of k-core.
Abstract: We study the random graph obtained by random deletion of vertices or edges from a random graph with given vertex degrees. A simple trick of exploding vertices instead of deleting them, enables us to derive results from known results for random graphs with given vertex degrees. This is used to study existence of giant component and existence of k-core. As a variation of the latter, we study also bootstrap percolation in random regular graphs. We obtain both simple new proofs of known results and new results. An interesting feature is that for some degree sequences, there are several or even infinitely many phase transitions for the k-core.
150 citations
TL;DR: In this article, the authors used the Malliavin calculus to obtain a new exact formula for the density of any random variable, which is measurable and differentiable with respect to a given isonormal Gaussian process.
Abstract: We show how to use the Malliavin calculus to obtain a new exact formula for the density $\rho$ of the law of any random variable $Z$ which is measurable and differentiable with respect to a given isonormal Gaussian process. The main advantage of this formula is that it does not refer to the divergence operator $\delta$ (dual of the Malliavin derivative $D$). The formula is based on an auxilliary random variable $G:= _H$, where $L$ is the generator of the so-called Ornstein-Uhlenbeck semigroup. The use of $G$ was first discovered by Nourdin and Peccati (PTRF 145 75-118 2009 MR-2520122 ), in the context of rates of convergence in law. Here, thanks to $G$, density lower bounds can be obtained in some instances. Among several examples, we provide an application to the (centered) maximum of a general Gaussian process. We also explain how to derive concentration inequalities for $Z$ in our framework.
123 citations
TL;DR: In this article, the authors define a dynamic model of random networks, where new vertices are connected to old ones with a probability proportional to a sublinear function of their degree.
Abstract: We define a dynamic model of random networks, where new vertices are connected to old ones with a probability proportional to a sublinear function of their degree. We first give a strong limit law for the empirical degree distribution, and then have a closer look at the temporal evolution of the degrees of individual vertices, which we describe in terms of large and moderate deviation principles. Using these results, we expose an interesting phase transition: in cases of strong preference of large degrees, eventually a single vertex emerges forever as vertex of maximal degree, whereas in cases of weak preference, the vertex of maximal degree is changing infinitely often. Loosely speaking, the transition between the two phases occurs in the case when a new edge is attached to an existing vertex with a probability proportional to the root of its current degree.
103 citations
TL;DR: In this article, the Harris-FKG inequality was extended to the more general setting of transient weighted graphs and it was shown that the critical value for the percolation of the vacant set is finite.
Abstract: In this article, we first extend the construction of random interlacements, introduced by A.S. Sznitman in [14], to the more general setting of transient weighted graphs. We prove the Harris-FKG inequality for this model and analyze some of its properties on specific classes of graphs. For the case of non-amenable graphs, we prove that the critical value $u_*$ for the percolation of the vacant set is finite. We also prove that, once $\mathcal{G}$ satisfies the isoperimetric inequality $I S_6$ (see (1.5)), $u_*$ is positive for the product $\mathcal{G} \times \mathbb{Z}$ (where we endow $\mathbb{Z}$ with unit weights). When the graph under consideration is a tree, we are able to characterize the vacant cluster containing some fixed point in terms of a Bernoulli independent percolation process. For the specific case of regular trees, we obtain an explicit formula for the critical value $u_*$.
95 citations
TL;DR: In this paper, the spectral gap of Markov chains constructed by parallel and simulated tempering is shown to be a function of the persistence property of the target distribution, which can lead unexpectedly to slow mixing that commonly used convergence diagnostics will fail to detect.
Abstract: We obtain upper bounds on the spectral gap of Markov chains constructed by parallel and simulated tempering, and provide a set of sufficient conditions for torpid mixing of both techniques. Combined with the results of Woodard, Schmidler and Huber (2009), these results yield a two-sided bound on the spectral gap of these algorithms. We identify a persistence property of the target distribution, and show that it can lead unexpectedly to slow mixing that commonly used convergence diagnostics will fail to detect. For a multimodal distribution, the persistence is a measure of how ``spiky'', or tall and narrow, one peak is relative to the other peaks of the distribution. We show that this persistence phenomenon can be used to explain the torpid mixing of parallel and simulated tempering on the ferromagnetic mean-field Potts model shown previously. We also illustrate how it causes torpid mixing of tempering on a mixture of normal distributions with unequal covariances in R^M, a previously unknown result with relevance to statistical inference problems. More generally, anytime a multimodal distribution includes both very narrow and very wide peaks of comparable probability mass, parallel and simulated tempering are shown to mix slowly.
76 citations
TL;DR: In this article, the random walk on supercritical percolation clusters was considered and a parabolic Harnack inequality, a local limit theorem and estimates on the Green's function were obtained.
Abstract: We consider the random walk on supercritical percolation clusters in $\mathbb{Z}^d$. Previous papers have obtained Gaussian heat kernel bounds, and a.s. invariance principles for this process. We show how this information leads to a parabolic Harnack inequality, a local limit theorem and estimates on the Green's function.
74 citations
TL;DR: In this article, the authors modify the usual Erdos-Renyi random graph evolution by letting connected clusters "burn down" (i.e., fall apart to disconnected single sites) due to a Poisson flow of lightnings.
Abstract: We modify the usual Erdos-Renyi random graph evolution by letting connected clusters 'burn down' (i.e. fall apart to disconnected single sites) due to a Poisson flow of lightnings. In a range of the intensity of rate of lightnings the system sticks to a permanent.
60 citations
TL;DR: The alpha-gamma trees as mentioned in this paper are a two-parameter family of discrete fragmentation trees that extends Ford's alpha model to multifurcating trees and includes the trees obtained by uniform sampling from Duquesne and Le Gall's stable continuum random tree.
Abstract: We introduce a simple tree growth process that gives rise to a new two-parameter family of discrete fragmentation trees that extends Ford's alpha model to multifurcating trees and includes the trees obtained by uniform sampling from Duquesne and Le Gall's stable continuum random tree. We call these new trees the alpha-gamma trees. In this paper, we obtain their splitting rules, dislocation measures both in ranked order and in size-biased order, and we study their limiting behaviour.
58 citations
TL;DR: In this article, the authors gave a new proof of a result of Kenyon that the growth exponent for loop-erased random walks in two dimensions is 5/4, using the convergence of LERW to Schramm-Loewner evolution with parameter 2.
Abstract: We give a new proof of a result of Kenyon that the growth exponent for loop-erased random walks in two dimensions is 5/4. The proof uses the convergence of LERW to Schramm-Loewner evolution with parameter 2, and is valid for irreducible bounded symmetric random walks on any two dimensional discrete lattice.
TL;DR: In this article, the asymptotic behavior of the least squares estimators of the unknown parameters of general pth-order bifurcating autoregressive processes is studied under very weak assumptions on the driven noise of the process.
Abstract: We study the asymptotic behavior of the least squares estimators of the unknown parameters of general pth-order bifurcating autoregressive processes. Under very weak assumptions on the driven noise of the process, namely conditional pair-wise independence and suitable moment conditions, we establish the almost sure convergence of our estimators together with the quadratic strong law and the central limit theorem. All our analysis relies on non-standard asymptotic results for martingales.
TL;DR: In this paper, the authors give three examples of stochastic processes in the Gelfand-Tsetlin cone in which each component evolves independently apart from a blocking and pushing interaction.
Abstract: We give three examples of stochastic processes in the Gelfand-Tsetlin cone in which each component evolves independently apart from a blocking and pushing interaction. These processes give rise to couplings between certain conditioned Markov processes, last passage times and exclusion processes. In the first two examples, we deduce known identities in distribution between such processes whilst in the third example, the components of the process cannot escape past a wall at the origin and we obtain a new relation.
TL;DR: This paper produces the first genuine sharp estimates in a context where entropy is important and shows how the potential theoretic approach can be used to prove sharp estimates on capacities and metastable exit times also in the case when the distribution of the random field is continuous.
Abstract: In this paper we study the metastable behavior of one of the simplest disordered spin system, the random field Curie-Weiss model. We will show how the potential theoretic approach can be used to prove sharp estimates on capacities and metastable exit times also in the case when the distribution of the random field is continuous. Previous work was restricted to the case when the random field takes only finitely many values, which allowed the reduction to a finite dimensional problem using lumping techniques. Here we produce the first genuine sharp estimates in a context where entropy is important.
TL;DR: In this paper, a log-type moment result for such random variables under minimal conditions is provided, which is then utilized for the study of related moments of a.i.d. martingales associated with the supercritical branching random walk.
Abstract: Infinite sums of i.i.d. random variables discounted by a multiplicative random walk are called perpetuities and have been studied by many authors. The present paper provides a log-type moment result for such random variables under minimal conditions which is then utilized for the study of related moments of a.s. limits of certain martingales associated with the supercritical branching random walk. The connection arises upon consideration of a size-biased version of the branching random walk originally introduced by Lyons. As a by-product, necessary and sufficient conditions for uniform integrability of these martingales are provided in the most general situation which particularly means that the classical (LlogL)-condition is not always needed.
TL;DR: In this paper, the authors derive Bernstein-like concentration inequalities for self-bounding functions using the Herbst argument, which involves comparison results between solutions of differential inequalities that may be interesting in their own right.
Abstract: We prove some new concentration inequalities for self-bounding functions using the entropy method. As an application, we recover Talagrand's convex distance inequality. The new Bernstein-like inequalities for self-bounding functions are derived thanks to a careful analysis of the so-called Herbst argument. The latter involves comparison results between solutions of differential inequalities that may be interesting in their own right.
TL;DR: In this paper, the inviscid shell model of turbulence has been shown to satisfy a LDP in the topology of uniform convergence on a set of variables, where the initial condition has to belong to $V$ and the proof is based on weak convergence of a family of stochastic control equations.
Abstract: LDP is proved for the inviscid shell model of turbulence. As the viscosity coefficient converges to 0 and the noise intensity is multiplied by its square root, we prove that some shell models of turbulence with a multiplicative stochastic perturbation driven by a $H$-valued Brownian motion satisfy a LDP in $\mathcal{C}([0,T],V)$ for the topology of uniform convergence on $[0,T]$, but where $V$ is endowed with a topology weaker than the natural one. The initial condition has to belong to $V$ and the proof is based on the weak convergence of a family of stochastic control equations. The rate function is described in terms of the solution to the inviscid equation.
TL;DR: In this article, the spectral empirical process of Wigner matrices under sixth-moment conditions, which is indexed by a set of functions with continuous fourth-order derivatives on an open interval including the support of the semicircle law, converges weakly in finite dimensions to a Gaussian process.
Abstract: In this paper, we prove that the spectral empirical process of Wigner matrices under sixth-moment conditions, which is indexed by a set of functions with continuous fourth-order derivatives on an open interval including the support of the semicircle law, converges weakly in finite dimensions to a Gaussian process.
TL;DR: In this paper, a comparison theorem and maximum principle for a local solution of quasi-linear parabolic stochastic PDEs were proved for Burgers type SPDEs, similar to the well known results in the deterministic case.
Abstract: We prove a comparison theorem and maximum principle for a local solution of quasi-linear parabolic stochastic PDEs, similar to the well known results in the deterministic case. The proofs are based on a version of Ito's formula and estimates for the positive part of a local solution which is non-positive on the lateral boundary. Moreover we shortly indicate how these results generalize for Burgers type SPDEs
TL;DR: In this paper, it was shown that the slope of the critical curve in the weak-disorder limit is strictly smaller than 1, which is the value given by the annealed inequality.
Abstract: For a much-studied model of random copolymer at a selective interface we prove that the slope of the critical curve in the weak-disorder limit is strictly smaller than 1, which is the value given by the annealed inequality. The proof is based on a coarse-graining procedure, combined with upper bounds on the fractional moments of the partition function.
TL;DR: In this paper, the authors considered the Dirichlet form and obtained upper and lower bounds on the heat kernel of the non-local part of a function in the nonlocal part, and proved a Harnack inequality and a regularity theorem for functions that are harmonic.
Abstract: We consider the Dirichlet form given by $$ {\cal E}(f,f) = \frac{1}{2}\int_{R^d}\sum_{i,j=1}^d a_{ij}(x)\frac{\partial f(x)}{\partial x_i} \frac{\partial f(x)}{\partial x_j} dx$$ $$ + \int_{R^d \times R^d} (f(y)-f(x))^2J(x,y)dxdy.$$ Under the assumption that the ${a_{ij}}$ are symmetric and uniformly elliptic and with suitable conditions on $J$, the nonlocal part, we obtain upper and lower bounds on the heat kernel of the Dirichlet form. We also prove a Harnack inequality and a regularity theorem for functions that are harmonic with respect to $\cal E$.
TL;DR: In this paper, a strong law of large numbers and a central limit theorem for the Parrondo player's sequence of profits were established, both in a one-parameter family of capital-dependent games and in a two-dimensional history-dependent game, with the potentially winning game being either a random mixture or a nonrandom pattern of the two losing games.
Abstract: That there exist two losing games that can be combined, either by random mixture or by nonrandom alternation, to form a winning game is known as Parrondo's paradox. We establish a strong law of large numbers and a central limit theorem for the Parrondo player's sequence of profits, both in a one-parameter family of capital-dependent games and in a two-parameter family of history-dependent games, with the potentially winning game being either a random mixture or a nonrandom pattern of the two losing games. We derive formulas for the mean and variance parameters of the central limit theorem in nearly all such scenarios; formulas for the mean permit an analysis of when the Parrondo effect is present.
TL;DR: In this paper, the authors give new exponential inequalities and Gaussian approximation results for sums of weakly dependent variables, leading to generalizations of Bernstein and Hoeffding inequalities, where an extra control term is added; this term contains conditional moments of the variables.
Abstract: We give new exponential inequalities and Gaussian approximation results for sums of weakly dependent variables. These results lead to generalizations of Bernstein and Hoeffding inequalities, where an extra control term is added; this term contains conditional moments of the variables.
TL;DR: In this article, the authors considered simple random walk on a discrete cylinder with base a large $d$-dimensional torus of side-length $N, and derived a lower bound on the disconnection time.
Abstract: We consider simple random walk on a discrete cylinder with base a large $d$-dimensional torus of side-length $N$, when $d$ is two or more. We develop a stochastic domination control on the local picture left by the random walk in boxes of side-length almost of order $N$, at certain random times comparable to the square of the number of sites in the base. We show a domination control in terms of the trace left in similar boxes by random interlacements in the infinite $(d+1)$-dimensional cubic lattice at a suitably adjusted level. As an application we derive a lower bound on the disconnection time of the discrete cylinder, which as a by-product shows the tightness of the laws of the ratio of the square of the number of sites in the base to the disconnection time. This fact had previously only been established when $d$ is at least 17.
TL;DR: In this paper, it was shown that quaternionic Gaussian random variables satisfy a generalization of the Wick formula for computing the expected value of products in terms of a family of graphical enumeration problems.
Abstract: We show that quaternionic Gaussian random variables satisfy a generalization of the Wick formula for computing the expected value of products in terms of a family of graphical enumeration problems. When applied to the quaternionic Wigner and Wishart families of random matrices the result gives the duality between moments of these families and the corresponding real Wigner and Wishart families.
TL;DR: In this paper, the authors derive a link between series expansions of Gaussian random vectors in a Banach space and Parseval frames in their reproducing kernel Hilbert space, and derive a new optimal expansion for fractional Ornstein-Uhlenbeck processes.
Abstract: We derive a precise link between series expansions of Gaussian random vectors in a Banach space and Parseval frames in their reproducing kernel Hilbert space. The results are applied to pathwise continuous Gaussian processes and a new optimal expansion for fractional Ornstein-Uhlenbeck processes is derived.
TL;DR: In this article, singular value techniques were developed in the context of time inhomogeneous finite Markov chains with the goal of obtaining quantitative results concerning the asymptotic behavior of such chains.
Abstract: We develop singular value techniques in the context of time inhomogeneous finite Markov chains with the goal of obtaining quantitative results concerning the asymptotic behavior of such chains. We introduce the notion of c-stability which can be viewed as a generalization of the case when a time inhomogeneous chain admits an invariant measure. We describe a number of examples where these techniques yield quantitative results concerning the merging of the distributions of the time inhomogeneous chain started at two arbitrary points.
TL;DR: In this paper, the authors constructed a martingale which has the same marginals as the arithmetic average of geometric Brownian motion in the convex order, and showed that the average is increasing with respect to the number of vertices.
Abstract: We construct a martingale which has the same marginals as the arithmetic average of geometric Brownian motion.This provides a short proof of the recent result due to P. Carr et al that the arithmetic average of geometric Brownian motion is increasing in the convex order. The Brownian sheet plays an essential role in the construction. Our method may also be applied when the Brownian motion is replaced by a stable subordinator.
TL;DR: In this article, it was shown that the Ito map is locally Lipschitz continuous in all its arguments and sufficient conditions for global existence for non-bounded vector fields.
Abstract: We prove that the Ito map, that is the map that gives the solution of a differential equation controlled by a rough path of finite $p$-variation with $p\in [2,3)$ is locally Lipschitz continuous in all its arguments and we give some sufficient conditions for global existence for non-bounded vector fields.
TL;DR: In this paper, a central limit theorem for the estimator of the union of balls centered at the X_i$ and with common radius $r_n was proved under mild assumptions on the probability density and the sequence of balls.
Abstract: Let $X_1,\ldots,X_n$ be $n$ independent observations drawn from a multivariate probability density $f$ with compact support $S_f$. This paper is devoted to the study of the estimator $\hat{S}_n$ of $S_f$ defined as the union of balls centered at the $X_i$ and with common radius $r_n$. Using tools from Riemannian geometry, and under mild assumptions on $f$ and the sequence $(r_n)$, we prove a central limit theorem for $\lambda (S_n \Delta S_f)$, where $\lambda$ denotes the Lebesgue measure on $\mathbb{R}^d$ and $\Delta$ the symmetric difference operation.