Showing papers in "ALEA-Latin American Journal of Probability and Mathematical Statistics in 2016"
TL;DR: In this paper, a piecewise deterministic Markov process is designed to sample the corresponding Gibbs measure, and an Eyring-Kramers formula is obtained for the exit time of the domain of a local minimum at low temperature, and a necessary and sufficient condition on the cooling schedule in a simulated annealing algorithm to ensure the process converges to the set of global minima.
Abstract: Given an energy potential on the Euclidian space, a piecewise deter- ministic Markov process is designed to sample the corresponding Gibbs measure. In dimension one an Eyring-Kramers formula is obtained for the exit time of the domain of a local minimum at low temperature, and a necessary and sufficient con- dition is given on the cooling schedule in a simulated annealing algorithm to ensure the process converges to the set of global minima. This condition is similar to the classical one for diffusions and involves the critical depth of the potential. In higher dimensions a non optimal sufficient condition is obtained.
42 citations
TL;DR: In this paper, a stochastic system of interacting particles is introduced, which is expected to furnish as the number of particles goes to infinity a 2-D Keller-Segel model.
Abstract: We introduce a stochastic system of interacting particles which is expected to furnish as the number of particles goes to infinity a stochastic approach of the 2-D Keller-Segel model. In this note, we prove existence and some uniqueness for the stochastic model for the parabolic-elliptic Keller-Segel equation, for all regimes under the critical mass. Prior results for existence and weak uniqueness have been very recently obtained by N. Fournier and B. Jourdain [6].
31 citations
TL;DR: In this paper, a bound on the total variation distance between a CMB distribution and the corresponding CMP limit is derived for the binomial binomial distribution, and convergence results and approximations are derived.
Abstract: The Conway-Maxwell-Poisson (CMP) distribution is a natural two-parameter generalisation of the Poisson distribution which has received some attention in the statis- tics literature in recent years by offering flexible generalisations of some well-known mod- els. In this work, we begin by establishing some properties of both the CMP distribution and an analogous generalisation of the binomial distribution, which we refer to as the CMB distribution. We also consider some convergence results and approximations, in- cluding a bound on the total variation distance between a CMB distribution and the corresponding CMP limit.
28 citations
TL;DR: In this article, a fractional counting process with jumps of amplitude 1,2,...,k, withk∈N, whose probabilistic ability to satisfy a suitablesystemoffractionaldifference-differential equations is considered.
Abstract: We consider a fractional counting process with jumps of amplitude 1,2,...,k, withk∈N, whoseprobabilitiessatisfy a suitablesystemoffractionaldifference-differential equations. We obtain the moment generating function and the probability law of the result- ing process in terms of generalized Mittag-Leffler functions. We also discuss two equiv- alent representations both in terms of a compound fractional Poisson process and of a subordinator governed by a suitable fractional Cauchy problem. The first occurrence time of a jump of fixed amplitude is proved to have the same distribution as the waiting time of the first event of a classical fractional Poisson process, this extending a well-known property of the Poisson process. When k = 2 we also express the distribution of the first passage time of the fractional counting process in an integral form. Finally, we show that the ratios given by the powers of the fractional Poisson process and of the countingprocess over their means tend to 1 in probability.
26 citations
TL;DR: In this paper, the central limit theorem for the magnetization rescaled by √ n of Ising models on random graphs was shown to hold in the whole region of the parameters β and B.
Abstract: The aim of this paper is to prove central limit theorems with respect to the annealed measure for the magnetization rescaled by √N of Ising models on random graphs. More precisely, we consider the general rank-1 inhomogeneous random graph (or generalized random graph), the 2-regular configuration model and the configuration model with degrees 1 and 2. For the generalized random graph, we first show the existence of a finite annealed inverse critical temperature 0≤ βan n 0 and B ≠ 0. In the case of the configuration model, the central limit theorem holds in the whole region of the parameters β and B, because phase transitions do not exist for these systems as they are closely related to one-dimensional Ising models. Our proofs are based on explicit computations that are possible since the Ising model on the generalized random graph in the annealed setting is reduced to an inhomogeneous Curie-Weiss model, while the analysis of the configuration model with degrees only taking values 1 and 2 relies on that of the classical one-dimensional Ising model.
25 citations
TL;DR: In this article, a new class of nonlinear Stochastic Differential Equations in the sense of McKean, related to non conservative nonlinear Partial Differential equations (PDEs), are introduced.
Abstract: We introduce a new class of nonlinear Stochastic Differential Equations in the sense of McKean, related to non conservative nonlinear Partial Differential equations (PDEs). We discuss existence and uniqueness pathwise and in law under various assumptions.
21 citations
TL;DR: In this paper, the authors consider a continuum percolation model on a set of independent Wiener sausages and derive moment estimates on the capacity of Wiener SAusages.
Abstract: We consider a continuum percolation model on $\R^d$, where $d\geq 4$.
The occupied set is given by
the union of independent Wiener sausages with radius $r$ running up to time $t$ and whose
initial points are distributed according to a homogeneous Poisson point process.
It was established in a previous work by Erhard, Mart\'{i}nez and Poisat~\cite{EMP13} that (1) if $r$ is small enough there is a non-trivial percolation transition
in $t$ occuring at a critical time $t_c(r)$ and (2) in the supercritical regime the unbounded cluster is unique. In this paper we investigate the asymptotic behaviour of the critical time when the radius $r$ converges to $0$. The latter does not seem to be deducible from simple scaling arguments. We prove that for $d\geq 4$, there is a positive constant $c$ such that
$c^{-1}\sqrt{\log(1/r)}\leq t_c(r)\leq c\sqrt{\log(1/r)}$ when $d=4$ and $c^{-1}r^{(4-d)/2}\leq t_c(r) \leq c\ r^{(4-d)/2}$ when $d\geq 5$, as $r$ converges to $0$. We derive along the way moment estimates on the capacity of Wiener sausages, which may be of independent interest.
14 citations
TL;DR: In this paper, the authors consider random walks indexed by arbitrary finite random or deterministic trees and derive a sufficient criterion which ensures that the maximal displacement of the tree-indexed random walk is determined by a single large jump.
Abstract: We consider random walks indexed by arbitrary finite random or deterministic trees. We derive a simple sufficient criterion which ensures that the maximal displacement of the tree-indexed random walk is determined by a single large jump. This criterion is given in terms of four quantities : the tail and the expectation of the random walk steps, the height of the tree and the number of its vertices. The results are applied to critical Galton--Watson trees with offspring distributions in the domain of attraction of a stable law.
13 citations
TL;DR: In this paper, it was shown that the block counting process is Siegmund dual to the fixation line of exchangeable coalescents and the associated limiting process is related to the frequencies of singletons of the coalescent.
Abstract: We study the block counting process and the fixation line of exchangeable coalescents. Formulas for the infinitesimal rates of both processes are provided. It is shown that the block counting process is Siegmund dual to the fixation line. For exchangeable coalescents restricted to a sample of size n and with dust we provide a convergence result for the block counting process as n tends to infinity. The associated limiting process is related to the frequencies of singletons of the coalescent. Via duality we obtain an analog convergence result for the fixation line of exchangeable coalescents with dust. The Dirichlet coalescent and the Poisson‐Dirichlet coalescent are studied in detail.
12 citations
TL;DR: In this paper, the authors considered self-avoiding walk on finite graphs with large girth and defined a critical exponent for sequences of graphs of size tending to infinity and showed that the expected length of a random self avoiding path is bounded.
Abstract: We consider self-avoiding walk on finite graphs with large girth. We study a few aspects of the model originally considered by Lawler, Schramm and Werner on finite balls in Z^d. The expected length of a random self avoiding path is considered. We also define a "critical exponent" $\gamma$ for sequences of graphs of size tending to infinity, and show that $\gamma = 1$ in the large girth case.
11 citations
TL;DR: In this article, it was shown that the third moment theorem holds for the case of normalized centered quadratic variations for stationary Gaussian sequences, and that convergence occurs if and only if the sequence's third moments tend to 0.
Abstract: In two new papers [2] and [8], sharp general quantitative bounds are given to complement the wellknown fourth moment theorem of Nualart and Peccati, by which a sequence in a …xed Wiener chaos converges to a normal law if and only if its fourth cumulant converges to 0. The bounds show that the speed of convergence is precisely of order the maximum of the fourth cumulant and the absolute value of the third moment (cumulant). Specializing to the case of normalized centered quadratic variations for stationary Gaussian sequences, we show that a third moment theorem holds: convergence occurs if and only if the sequence’s third moments tend to 0. This is proved for sequences with general decreasing covariance, by using the result of [8], and …nding the exact speed of convergence to 0 of the quadratic variation’s third and fourth cumulants. [8] also allows us to derive quantitative estimates for the speeds of convergence in a class of log-modulated covariance structures, which puts in perspective the notion of critical Hurst parameter when studying the convergence of fractional Brownian motion’s quadratic variation. We also study the speed of convergence when the limit is not Gaussian but rather a second-Wiener-chaos law. Using a log-modulated class of spectral densities, we recover a classical result of Dobrushin-Major/Taqqu whereby the limit is a Rosenblatt law, and we provide new convergence speeds. The conclusion in this case is that the price to pay to obtain a Rosenblatt limit despite a slowly varying modulation is a very slow convergence speed, roughly of the same order as the modulation.
TL;DR: In this article, it was shown that conditioning a multitype Galton-Watson process on having an infinite total progeny leads to a process presenting the features of a Q-process.
Abstract: Conditioning a multitype Galton-Watson process to stay alive into the indefinite future leads to what is known as its associated Q-process. We show that the same holds true if the process is conditioned to reach a positive threshold or a non-absorbing state. We also demonstrate that the stationary measure of the Q-process, obtained by construction as two successive limits (first by delaying the extinction in the original process and next by considering the long-time behavior of the obtained Q-process), is as a matter of fact a double limit. Finally, we prove that conditioning a multitype branching process on having an infinite total progeny leads to a process presenting the features of a Q-process. It does not however coincide with the original associated Q-process, except in the critical regime.
TL;DR: In this paper, the authors extend the investigation of the Fleming-Viot process in discrete space to two specific examples, i.e., a random walk on the complete graph and a Markov chain in a two state space.
Abstract: The purpose of this paper is to extend the investigation of the Fleming-Viot process in discrete space started in a previous work to two specific examples. The first one corresponds to a random walk on the complete graph. Due to its geometry, we establish several explicit and optimal formulas for the Fleming-Viot process (invariant distribution, correlations, spectral gap). The second example corresponds to a Markov chain in a two state space. In this case, the study of the Fleming-Viot particle system is reduced to the study of birth and death process with quadratic rates.
TL;DR: In this paper, the authors studied the interface of the symmetric multitype contact process on Z and proved that the position of the interface converges to Brownian motion under diffusive scaling.
Abstract: We study the interface of the symmetric multitype contact process on Z. In this process, each site of Z is either empty or occupied by an individual of one of two species. Each individual dies with rate 1 and attempts to give birth with rate 2R lambda; the position for the possible new individual is chosen uniformly at random within distance R of the parent, and the birth is suppressed if this position is already occupied. We consider the process started from the configuration in which all sites to the left of the origin are occupied by one of the species and all sites to the right of the origin by the other species, and study the evolution of the region of interface between the two species. We prove that, under diffusive scaling, the position of the interface converges to Brownian motion.
Journal Article•
TL;DR: In this article, the rescaled maximum of the discrete Gaussian Free Field (DGFF) in dimension larger or equal to 3 is in the maximal domain of attraction of the Gumbel distribution.
Abstract: We show that the rescaled maximum of the discrete Gaussian Free Field (DGFF) in dimension larger or equal to 3 is in the maximal domain of attraction of the Gumbel distribution. The result holds both for the infinite-volume field as well as the field with zero boundary conditions. We show that these results follow from an interesting application of the Stein-Chen method from Arratia et al. (1989).
TL;DR: In this article, the authors introduced two stationary versions of two discrete variants of Hammersley's process in a finite box, which allowed them to recover in a unified and simple way the laws of large numbers proved by T. Seppalainen for two generalized Ulam's problems.
Abstract: We introduce two stationary versions of two discrete variants of Hammersley's process in a finite box, this allows us to recover in a unified and simple way the laws of large numbers proved by T. Seppalainen for two generalized Ulam's problems. As a by-product we obtain an elementary solution for the original Ulam problem. We also prove that for the first process defined on Z, Bernoulli product measures are the only extremal and translation-invariant stationary measures.
TL;DR: Goldstein and Reinert as mentioned in this paper showed that the distance between the scaled number of white balls drawn in a P\'olya-Eggenberger urn and its limiting distribution is of order $1/n^\delta.
Abstract: A consequence of de Finetti's representation theorem is that for every infinite sequence of exchangeable 0-1 random variables $(X_k)_{k\geq1}$, there exists a probability measure $\mu$ on the Borel sets of $[0,1]$ such that $\bar X_n = n^{-1} \sum_{i=1}^n X_i$ converges weakly to $\mu$. For a wide class of probability measures $\mu$ having smooth density on $(0,1)$, we give bounds of order $1/n$ with explicit constants for the Wasserstein distance between the law of $\bar X_n$ and $\mu$. This extends a recent result {by} Goldstein and Reinert \cite{goldstein2013stein} regarding the distance between the scaled number of white balls drawn in a P\'olya-Eggenberger urn and its limiting distribution. We prove also that, in the most general cases, the distance between the law of $\bar X_n$ and $\mu$ is bounded below by $1/n$ and above by $1/\sqrt{n}$ (up to some multiplicative constants). For every $\delta \in [1/2,1]$, we give an example of an exchangeable sequence such that this distance is of order $1/n^\delta$.
TL;DR: In this article, the authors apply the theory of Lyapunov functions and find explicit estimates for the rate of exponential convergence to the stationary distribution, as time goes to infinity.
Abstract: Consider a reflected jump-diffusion on the positive half-line. Assume it is stochastically ordered. We apply the theory of Lyapunov functions and find explicit estimates for the rate of exponential convergence to the stationary distribution, as time goes to infinity. This continues the work of Lund, Meyn and Tweedie (1996). We apply these results to systems of two competing Levy particles with rank-dependent dynamics.
TL;DR: In this article, a random field constructed from the two-dimensional Gaussian free field (GFF) by modifying the variance along the scales in the neighborhood of each point was studied.
Abstract: In this paper, we study a random field constructed from the two-dimensional Gaussian free field (GFF) by modifying the variance along the scales in the neighborhood of each point. The construction can be seen as a local martingale transform and is akin to the time-inhomogeneous branching random walk. In the case where the variance takes finitely many values, we compute the first order of the maximum and the log-number of high points. These quantities were obtained by Bolthausen, Deuschel and Giacomin (2001) and Daviaud (2006) when the variance is constant on all scales. The proof relies on a truncated second moment method proposed by Kistler (2015), which streamlines the proof of the previous results. We also discuss possible extensions of the construction to the continuous GFF.
TL;DR: In this paper, a general type of a split-break process with Gaussian innovations (henceforth, Gaussian Split-Break or GSB process) is considered.
Abstract: A general type of a Split-BREAK process with Gaussian innovations (henceforth, Gaussian Split-BREAK or GSB process) is considered. The basic stochastic properties of the model are studied and its characteristic function derived. A procedure to estimate the parameter of the GSB model based on the Empirical Characteristic Function (ECF) is proposed. Our simulations suggest that the proposed method performs well compared to a Method of Moment procedure used as benchmark. The empirical use of the GSB model is illustrated with an application to the time series of total values of shares traded at Belgrade Stock Exchange.
TL;DR: The main purpose of this work is to study the existence of a phase transition between extinction and persistence of the infection in the parameter region where the hosts survive.
Abstract: The stacked contact process is a stochastic model for the spread of an infection within a population of hosts located on the d-dimensional integer lattice. Regardless of whether they are healthy or infected, hosts give birth and die at the same rate and in accordance to the evolution rules of the neutral multitype contact process. The infection is transmitted both vertically from infected parents to their offspring and horizontally from infected hosts to nearby healthy hosts. The population survives if and only if the common birth rate of healthy and infected hosts exceeds the critical value of the basic contact process. The main purpose of this work is to study the existence of a phase transition between extinction and persistence of the infection in the parameter region where the hosts survive.
TL;DR: In this paper, the persistence properties of random processes in Brownian scenery were studied, which are examples of non-Markovian and non-Gaussian processes, and the asymptotic behavior for large quantities of time was studied.
Abstract: In this paper we consider the persistence properties of random processes in Brownian scenery, which are examples of non-Markovian andnon-Gaussian processes. More precisely we study the asymptotic behaviour for large $T$, of the probability $P[ \sup_{t\in[0,T]} \Delta_t \leq 1] $where $\Delta_t = \int_{\mathbb{R}} L_t(x) \, dW(x).$Here $W={W(x); x\in\mathbb{R}}$ is a two-sided standard real Brownian motion and ${L_t(x); x\in\mathbb{R},t\geq 0}$ isthe local time of some self-similar random process $Y$, independent from the process $W$. We thus generalize the results of \cite{BFFN} where the increments of $Y$ were assumed to be independent.
TL;DR: In this article, the authors investigated whether convergence of the third and fourth moments of such a suitably normalized sequence to a centred Gamma law implies convergence in distribution of the involved random variables.
Abstract: This paper deals with sequences of random variables belonging to a
fixed chaos of order q generated by a Poisson random measure on a Polish space. The problem is investigated whether convergence of the third and fourth moment of such a suitably normalized sequence to the third and fourth moment of a centred Gamma law implies convergence in distribution of the involved random variables. A positive answer is obtained for q = 2 and q = 4. The proof of this four moments theorem is based on a number of new estimates for contraction norms. Applications concern homogeneous sums and U-statistics on the Poisson space.
TL;DR: In this article, Duquesne and Wang introduced weights on the unrooted unlabeled plane trees as follows: for each p ≥ 1, let μp be a probability measure on the set of nonnegative integers whose mean is bounded by 1; then the μp-weight of a plane tree t is defined as Π μ_p(degree(v) − 1), where the product is over the vertices v of t.
Abstract: We introduce weights on the unrooted unlabelled plane trees as follows: for each p ≥ 1, let μp be a probability measure on the set of nonnegative integers whose mean is bounded by 1; then the μp-weight of a plane tree t is defined as Π μ_p(degree(v) − 1), where the product is over the set of vertices v of t. We study the random plane tree T_p which has a fixed diameter p and is sampled according to probabilities proportional to these μ_p-weights. We prove that, under the assumption that the sequence of laws μ_p, p ≥ 1, belongs to the domain of attraction of an infinitely divisible law, the scaling limits of (T_p , p ≥ 1) are random compact real trees called the unrooted Levy trees, which have been introduced in Duquesne and Wang (2016+).
TL;DR: In this paper, the authors studied the limit law of the supremum of the local time, as well as the position of the favorite sites of a one-dimensional diffusion in a drifted Brownian potential.
Abstract: We study a one-dimensional diffusion $X$ in a drifted Brownian potential $W_\kappa$, with $ 0 0$.
In particular we characterize the limit law of the supremum of the local time, as well as the position of the favorite sites. These limits can be written explicitly from a two dimensional stable Levy process. Our analysis is based on the study of an extension of the renewal structure which is deeply involved in the asymptotic behavior of $X$.
TL;DR: In this article, the authors established pathwise functional Itô formulas for nonsmooth functionals of real-valued continuous semimartingales under finite (p, q)variation regularity assumptions in the sense of Young integration theory.
Abstract: In this work, we establish pathwise functional Itô formulas for nonsmooth functionals of real-valued continuous semimartingales. Under finite (p, q)variation regularity assumptions in the sense of two-dimensional Young integration theory, we establish a pathwise local-time decomposition Ft(Xt) = F0(X0) + ∫ t
TL;DR: In this article, a model for email communication due to Gabrielli and Caldarelli was studied, where the receiver assigns i.i.d. priorities to incoming emails according to some atomless law and always answers the email in the mailbox with the highest priority.
Abstract: We study a model for email communication due to Gabrielli and Caldarelli, where someone receives and answers emails at the times of independent Poisson processes with intensities λin > λout. The receiver assigns i.i.d. priorities to incoming emails according to some atomless law and always answers the email in the mailbox with the highest priority. Since the frequency of incoming emails is higher than the frequency of answering, below a critical priority, the mailbox fills up ad infinitum. We prove a theorem about the limiting shape of the mailbox just above the critical point, linking it to the convex hull of Brownian motion. We conjecture that this limiting shape is universal in a class of similar models, including a model for the evolution of an order book due to Stigler and Luckock.
TL;DR: In this article, it was shown that all geodesic rays in the uniform infinite half-planar quadrangulation intersect the boundary infinitely many times, answering a recent question of Curien.
Abstract: We show that all geodesic rays in the uniform infinite half-planar quadrangulation (UIHPQ) intersect the boundary infinitely many times, answering thereby a recent question of Curien. However, the possible intersection points are sparsely distributed along the boundary. As an intermediate step, we show that geodesic rays in the UIHPQ are proper, a fact that was recently established by Caraceni and Curien by a reasoning different from ours. Finally, we argue that geodesic rays in the uniform infinite half-planar triangulation behave in a very similar manner, even in a strong quantitative sense.
TL;DR: In this article, the authors considered a weighted random walk on the backbone of an oriented percolation cluster and determined necessary conditions on the weights for Brownian scaling limits under the annealed and quenched law.
Abstract: . We consider a weighted random walk on the backbone of an oriented percolation cluster. We determine necessary conditions on the weights for Brownian scaling limits under the annealed and the quenched law. This model is a random walk in dynamic random environment (RWDRE), where the environment is mixing, non-Markovian and not elliptic. We provide a generalization of results obtained previously by Birkner et al. (2013).