Showing papers in "ALEA-Latin American Journal of Probability and Mathematical Statistics in 2019"
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TL;DR: In this paper, the authors established limit theorems for the log-volume and the volume of the convex hull of the simplex in high dimensions, and showed that the fluctuations of the volume are normal (respectively, log-normal) if $r=o(n)$ and $r\sim \alpha n$ for some δ < 1.
Abstract: Let $r=r(n)$ be a sequence of integers such that $r\leq n$ and let $X_1,\ldots,X_{r+1}$ be independent random points distributed according to the Gaussian, the Beta or the spherical distribution on $\mathbb{R}^n$. Limit theorems for the log-volume and the volume of the random convex hull of $X_1,\ldots,X_{r+1}$ are established in high dimensions, that is, as $r$ and $n$ tend to infinity simultaneously. This includes, Berry-Esseen-type central limit theorems, log-normal limit theorems, moderate and large deviations. Also different types of mod-$\phi$ convergence are derived. The results heavily depend on the asymptotic growth of $r$ relative to $n$. For example, we prove that the fluctuations of the volume of the simplex are normal (respectively, log-normal) if $r=o(n)$ (respectively, $r\sim \alpha n$ for some $0 < \alpha < 1$).
50 citations
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23 citations
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TL;DR: In this article, the authors initiated the study of generalized divide and color models, a very interesting special case of this is the "Divide and Color Model" introduced and studied by Olle Haggstrom.
Abstract: In this paper, we initiate the study of "Generalized Divide and Color Models". A very interesting special case of this is the "Divide and Color Model" (which motivates the name we use) introduced and studied by Olle Haggstrom. In this generalized model, one starts with a finite or countable set V, a random partition of V and a parameter p ∈ [0; 1]. The corresponding Generalized Divide and Color Model is the [0; 1]-valued process indexed by V obtained by independently, for each partition element in the random partition chosen, with probability p, assigning all the elements of the partition element the value 1, and with probability 1 - p, assigning all the elements of the partition element the value 0. Some of the questions which we study here are the following. Under what situations can different random partitions give rise to the same color process? What can one say concerning exchangeable random partitions? What is the set of product measures that a color process stochastically dominates? For random partitions which are translation invariant, what ergodic properties do the resulting color processes have? The motivation for studying these processes is twofold; on the one hand, we believe that this is a very natural and interesting class of processes that deserves investigation and on the other hand, a number of quite varied well-studied processes actually fall into this class such as (1) the Ising model, (2) the fuzzy Potts model, (3) the stationary distributions for the Voter Model, (4) random walk in random scenery and of course (5) the original Divide and Color Model.
15 citations
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TL;DR: In this paper, the Bernstein and Hoeffding type inequalities for regenerative Markov chains were presented and generalized to the case of Markov chain, and exponential bounds for suprema of empirical processes over a class of functions whose size is controlled by its uniform entropy number were established.
Abstract: The purpose of this paper is to present Bernstein and Hoeffding type inequalities for regenerative Markov chains. Furthermore, we
generalize these results and establish exponential bounds for suprema of empirical processes over a class of functions F which size is controlled by its
uniform entropy number. All constants involved in the bounds of the considered inequalities are given in an explicit form which can be advantageous
for practical considerations. We present the theory for regenerative Markov chains, however the inequalities are also valid in the Harris recurrent case.
14 citations
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13 citations
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TL;DR: In this article, the authors improved the bound to $.2870 for the variant in which particles start at the integers and showed that the survival of a particle can be viewed as a renewal property of the Galton-Watson process.
Abstract: Three-speed ballistic annihilation starts with infinitely many particles on the real line. Each is independently assigned either speed-$0$ with probability $p$, or speed-$\pm 1$ symmetrically with the remaining probability. All particles simultaneously begin moving at their assigned speeds and mutually annihilate upon colliding. Physicists conjecture when $p \leq p_c = 1/4$ all particles are eventually annihilated. Dygert et. al. prove $p_c \leq .3313$, while Sidoravicius and Tournier describe an approach to prove $p_c \leq .3281$. For the variant in which particles start at the integers, we improve the bound to $.2870$. A renewal property lets us equate survival of a particle to the survival of a Galton-Watson process whose offspring distribution a computer can rigorously approximate. This approach may help answer the nearly thirty-year old conjecture that $p_c >0$.
12 citations
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TL;DR: In this paper, the concept of Levy subordinator (non-decreasing paths, infinitely divisible (ID)law at any point in time) is generalized to a family of non-decreeasing stochastic processes, which are parameterized in terms of two Bernstein functions.
Abstract: The concept of a Levy subordinator (non-decreasing paths, infinitely divisible (ID)
law at any point in time) is generalized to a family of non-decreasing stochastic processes
which are parameterized in terms of two Bernstein functions. Whereas the
independent increments property is only maintained in the Levy subordinator special
case, the considered family is always strongly infinitely divisible with respect to
time (IDT), meaning that a path can be represented in distribution as a finite sum
with arbitrarily many summands of independent and identically distributed paths
of another process. Besides distributional properties of the process, we present two
applications to the design of accurate and efficient simulation algorithms, emphasizing
our interest in the investigated processes. First, each member of the considered
family corresponds uniquely to an exchangeable max-stable sequence of random
variables, and we demonstrate how the associated extreme-value copula can be simulated
exactly and effciently from its Pickands dependence measure. Second, we
show how one obtains different series and integral representations for infinitely divisible
probability laws by varying the parameterizing pair of Bernstein functions,
without changing the one-dimensional law of the process. As a particular example,
we present an exact simulation algorithm for compound Poisson distributions from
the Bondesson class, for which the generalized inverse of the distribution function
of the associated Stieltjes measure can be evaluated accurately.
11 citations
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9 citations
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TL;DR: In this paper, the authors studied the edge statistics of the Gaussian ensemble and proved that the associated extreme point process converges in distribution to a Poisson point process as the inverse temperature tends to zero as the number of particles tends to infinity.
Abstract: We study the asymptotic edge statistics of the Gaussian $\beta$-ensemble, a collection of $n$ particles, as the inverse temperature $\beta$ tends to zero as $n$ tends to infinity. In a certain decay regime of $\beta$, the associated extreme point process is proved to converge in distribution to a Poisson point process as $n\to +\infty$. We also extend a well known result on Poisson limit for Gaussian extremes by showing the existence of an edge regime that we did not find in the literature.
9 citations
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TL;DR: In this article, the authors consider random graphs with uniformly bounded edges on a Poisson point process conditioned to contain the origin and prove that in the subcritical phase the probability that the origin is connected to some point at distance $n$ decays exponentially in
Abstract: We consider random graphs with uniformly bounded edges on a Poisson point process conditioned to contain the origin. In particular we focus on the random connection model, the Boolean model and Miller-Abrahams random resistor network with lower-bounded conductances. The latter is relevant for the analysis of conductivity by Mott variable range hopping in strongly disordered systems. By using the method of randomized algorithms developed by Duminil-Copin et al. we prove that in the subcritical phase the probability that the origin is connected to some point at distance $n$ decays exponentially in $n$, while in the supercritical phase the probability that the origin is connected to infinity is strictly positive and bounded from below by a term proportional to $ (\\lambda-\\lambda_c)$, $\\lambda$ being the density of the Poisson point process and $\\lambda_c$ being the critical density.
9 citations
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TL;DR: In this paper, a probabilistic approach to the core-size in random maps is presented, which yields straightforward and singularity analysis-free proofs of some results of Banderier, Flajolet, Schaeffer and Soria, and also yields convergence in distribution of the largest 2-connected block in a large random map, for any fixed k > 1, to a Fr\'echet-type extreme order statistic.
Abstract: We present a probabilistic approach to the core-size in random maps, which yields straightforward and singularity analysis-free proofs of some results of Banderier, Flajolet, Schaeffer and Soria. The proof also yields convergence in distribution of the rescaled size of the k'th largest 2-connected block in a large random map, for any fixed k > 1, to a Fr\'echet-type extreme order statistic. This seems to be a new result even when k=2.
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TL;DR: In this article, the authors study the dynamics of a nonlinear dissipative reaction-diffusion equation with well-separated stable states which is perturbed by multiplicative L\'evy noise with a regularly varying component at intensity.
Abstract: This article studies the dynamics of a nonlinear dissipative reaction-diffusion equation with well-separated stable states which is perturbed by infinite-dimensional multiplicative L\'evy noise with a regularly varying component at intensity $\epsilon>0$. The main results establish the precise asymptotics of the first exit times and locus of the solution $X^\epsilon$ from the domain of attraction of a deterministic stable state, in the limit as $\epsilon\rightarrow 0$. In contrast to the exponential growth for respective Gaussian perturbations the exit times grow essentially as a power function of the noise intensity as $\epsilon \rightarrow 0$ with the exponent given as the tail index $-\alpha$, $\alpha>0,$ of the L\'evy measure, analogously to the case of additive noise in Debussche et al (2013). In this article we substantially improve their quadratic estimate of the small jump dynamics and derive a new exponential estimate of the stochastic convolution for stochastic L\'evy integrals with bounded jumps based on the recent pathwise Burkholder-Davis-Gundy inequality by Siorpaes (2018). This allows to cover perturbations with general tail index $\alpha>0$, multiplicative noise and perturbations of the linear heat equation. In addition, our convergence results are probabilistically strongest possible. Finally, we infer the metastable convergence of the process on the common time scale $t/\epsilon^\alpha$ to a Markov chain switching between the stable states of the deterministic dynamical system.
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TL;DR: This work presents a Gaussian process that arise from the iteration of p fractional Ornstein-Uhlenbeck processes generated by the same fractional Brownian motion, and proves that it results in a short memory processes.
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TL;DR: In this paper, the authors prove local limit theorems for mod-ϕ convergent sequences of random variables, ϕ being a stable distribution, and identify the infinitesimal scales at which the stable approximation is valid.
Abstract: We prove local limit theorems for mod-ϕ convergent sequences of random variables, ϕ being a stable distribution. In particular, we give two new proofs of the local limit theorem stated in Delbaen et al. (2015): one proof based on the notion of zone of control introduced in Feray et al. (2019+a), and one proof based on the notion of mod-ϕ convergence in $\textit{L}^1$$(i\mathbb{R})$. These new approaches allow us to identify the infinitesimal scales at which the stable approximation is valid. We complete our analysis with a large variety of examples to which our results apply, and which stem from random matrix theory, number theory, combinatorics or statistical mechanics.
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TL;DR: In this article, the authors studied the asymptotic properties of nearest-neighbor random walks in 1d random environment under the influence of an external field of intensity λ ∈ R. They showed that the limiting velocity v(λ) is always increasing and that it is everywhere analytic except at most in two points λ− and λ+.
Abstract: We study the asymptotic properties of nearest-neighbor random walks in 1d random environment under the influence of an external field of intensity λ ∈ R. For ergodic shift-invariant environments, we show that the limiting velocity v(λ) is always increasing and that it is everywhere analytic except at most in two points λ− and λ+. When λ− and λ+ are distinct, v(λ) might fail to be continuous. We refine the assumptions in [?] for having a recentered CLT with diffusivity σ 2 (λ) and give explicit conditions for σ 2 (λ) to be analytic. For the random conductance model we show that, in contrast with the deterministic case, σ 2 (λ) is not monotone on the positive (resp. negative) half-line and that it is not differentiable at λ = 0. For this model we also prove the Einstein Relation, both in discrete and continuous time, extending the result of Lam and Depaw (2016).
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TL;DR: In this article, it was shown that the Fleming-Viot process selects the minimal quasi-stationary distribution for Markov processes with soft killing on non-compact state spaces.
Abstract: We prove under mild conditions that the Fleming-Viot process selects the minimal quasi-stationary distribution for Markov processes with soft killing on non-compact state spaces. Our results are applied to multi-dimensional birth and death processes, continuous time Galton-Watson processes and diffusion processes with soft killing.
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TL;DR: In this article, the authors studied functional convergence of sums of moving averages with random coefficients and heavy-tailed innovations under some standard moment conditions and the assumption that all partial sums of the series of coefficients are a.s.
Abstract: We study functional convergence of sums of moving averages with random coefficients and heavy-tailed innovations. Under some standard moment conditions and the assumption that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series we obtain functional convergence of the corresponding partial sum stochastic process in the space $D[0,1]$ of cadlag functions with the Skorohod $M_{2}$ topology.
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TL;DR: In this paper, a two-dimensional distribution-valued quadratic field associated with the one-dimensional Ornstein-Uhlenbeck process was introduced, and it was shown that the stationary quadrastic fluctuations of the simple exclusion process, in diffusive scaling, converge to this field.
Abstract: We introduce a two-dimensional, distribution-valued field, which we call the quadratic field, associated with the one-dimensional Ornstein-Uhlenbeck process and we prove that the stationary quadratic fluctuations of the simple exclusion process, in the diffusive scaling, converge to this quadratic field. Moreover, we prove that this quadratic field evaluated at the diagonal corresponds to the Wick-renormalized square of the Ornstein-Uhlenbeck process, and we use this new representation in order to prove some small and large-time properties of it.
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TL;DR: In this paper, the authors consider the backbone of the infinite cluster generated by supercritical oriented site percolation in dimension 1 + 1 and show that, after diffusive rescaling, the collection of paths converges in distribution to the Brownian web.
Abstract: We consider the backbone of the infinite cluster generated by supercritical oriented site percolation in dimension 1 +1. A directed random walk on this backbone can be seen as an "ancestral line" of an individual sampled in the stationary discrete-time contact process. Such ancestral lineages were investigated in [BCDG13] where a central limit theorem for a single walker was proved. Here, we consider infinitely many coalescing walkers on the same backbone starting at each space-time point. We show that, after diffusive rescaling, the collection of paths converges in distribution to the Brownian web. Hence, we prove convergence to the Brownian web for a particular system of coalescing random walks in a dynamical random environment. An important tool in the proof is a tail bound on the meeting time of two walkers on the backbone, started at the same time. Our result can be interpreted as an averaging statement about the percolation cluster: apart from a change of variance, it behaves as the full lattice, i.e. the effect of the "holes" in the cluster vanishes on a large scale.
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TL;DR: In this article, the authors investigate a probabilistic model for routeing of messages in relay-augmented multihop ad-hoc networks, where each transmitter sends one message to the origin.
Abstract: We investigate a probabilistic model for routeing of messages in relay-augmented multihop ad-hoc networks, where each transmitter sends one message to the origin. Given the (random) transmitter locations, we weight the family of random, uniformly distributed message trajectories by an exponential probability weight, favouring trajectories with low interference (measured in terms of signal-to-interference ratio) and trajectory families with little congestion (measured in terms of the number of pairs of hops using the same relay). Under the resulting Gibbs measure, the system targets the best compromise between entropy, interference and congestion for a common welfare, instead of an optimization of the individual trajectories.
In the limit of high spatial density of users, we describe the totality of all the message trajectories in terms of empirical measures. Employing large deviations arguments, we derive a characteristic variational formula for the limiting free energy and analyse the minimizer(s) of the formula, which describe the most likely shapes of the trajectory flow. The empirical measures of the message trajectories well describe the interference, but not the congestion; the latter requires introducing an additional empirical measure. Our results remain valid under replacing the two penalization terms by more general functionals of these two empirical measures.
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TL;DR: In this article, the eigenvalues of the Laguerre process and certain distinguished dynamics on partitions are analyzed and the relation between the eigvalues and the dynamics on the partitions is established.
Abstract: Probability measures and stochastic dynamics on matrices and on partitions are related by standard, albeit technical, discrete to continuous scaling limits. In this paper we provide exact relations, that go in both directions, between the eigenvalues of the Laguerre process and certain distinguished dynamics on partitions. This is done by generalizing to the multidimensional setting recent results of Miclo and Patie on linear one-dimensional diffusions and birth and death chains. As a corollary, we obtain an exact relation between the Laguerre and Meixner ensembles. Finally, we explain the deep connections with the Young bouquet and the z-measures on partitions.
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TL;DR: In this paper, it was shown that every adapted Brownian bridge on a geodesically complete connected Riemannian manifold is a semimartingale including its terminal time, without any further assumptions on the geometry.
Abstract: I prove that every adapted Brownian bridge on a geodesically complete connected Riemannian manifold is a semimartingale including its terminal time, without any further assumptions on the geometry. In particular, it follows that every such process can be horizontally lifted to a smooth principal fiber bundle with connection, including its terminal time. The proof is based on a localized Hamilton-type gradient estimate by Arnaudon/Thalmaier.
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TL;DR: In this article, the authors present an elementary approach to the order of fluctuations for the free energy in the Sherrington-Kirkpatrick mean field spin glass model at and near the critical temperature.
Abstract: We present an elementary approach to the order of fluctuations for the free energy in the Sherrington-Kirkpatrick mean field spin glass model at and near the critical temperature. It is proved that at the critical temperature the variance of the free energy is of $O((\log N)^2).$ In addition, we show that if one approaches the critical temperature from the low temperature regime at the rate $O(N^{-\alpha})$ for some $\alpha>0,$ then the variance is of $O((\log N)^2+N^{1-\alpha}).$
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TL;DR: In this article, the authors consider a variant of the continuous and discrete Ulam-Hammersley problems, where the maximal length of an increasing path through a Poisson point process (or a Bernoulli point process) with the restriction that there must be minimal gaps between abscissae and ordinates of successive points of the path is studied.
Abstract: We consider a variant of the continuous and discrete Ulam-Hammersley problems: we study the maximal length of an increasing path through a Poisson point process (or a Bernoulli point process) with the restriction that there must be minimal gaps between abscissae and ordinates of successive points of the path.
For both cases (continuous and discrete) our approach rely on couplings with well-studied models: respectively the classical Ulam-Hammersley problem and last-passage percolation with geometric weights. Thanks to these couplings we obtain explicit limiting shapes in both settings.
We also establish that, as in the classical Ulam-Hammersley problem, the fluctuations around the mean are given by the Tracy-Widom distribution.
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TL;DR: In this paper, a family of real random variables arising from the supersymmetric nonlinear sigma model and containing the family β$ introduced by Sabot, Tarr\`es, and Zeng [STZ17] in the context of the vertex-reinforced jump process was introduced.
Abstract: We introduce a family of real random variables $(\beta,\theta)$ arising from the supersymmetric nonlinear sigma model and containing the family $\beta$ introduced by Sabot, Tarr\`es, and Zeng [STZ17] in the context of the vertex-reinforced jump process. Using this family we construct an exponential martingale generalizing the one considered in [DMR17]. Moreover, using the full supersymmetric nonlinear sigma model we also construct a generalization of the exponential martingale involving Grassmann variables.
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TL;DR: This paper aims at comparing theoretical approximations of the tail of the maximum of stochastic processes and the corresponding numerical evaluations of the Pickands or double sum method, the Rice Method, the Euler Characteristic method and a new one called the Poisson method.
Abstract: This paper aims at comparing theoretical approximations of the tail of the maximum of stochastic processes and the corresponding numerical evaluations. More particularly, we focus on the Pickands or double sum method, the Rice method, the Euler Characteristic method and a new one called the Poisson method. The numerical evaluation, performed using mainly Quasi Monte-Carlo integration and adaptations of the programs of Genz, show the domains of validity of each method.
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TL;DR: This paper is devoted to the estimation of the relative scale parameter of conditioned Galton-Watson trees, where new estimators are introduced and their consistency is stated.
Abstract: Tree-structured data naturally appear in various fields, particularly in biology where plants and blood vessels may be described by trees, but also in computer science because XML documents form a tree structure. This paper is devoted to the estimation of the relative scale of ordered trees that share the same layout. The theoretical study is achieved for the stochastic model of conditioned Galton-Watson trees. New estimators are introduced and their consistency is stated. A comparison is made with an existing approach of the literature. A simulation study shows the good behavior of our procedure on finite-sample sizes and from missing or noisy data. An application to the analysis of revisions of Wikipedia articles is also considered through real data.
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TL;DR: In this paper, the authors investigate the problem of constructing a Markov process on edge-spin configurations which realizes a coupling between a Glauber dynamics of the Ising model and a dynamical evolution of the percolation configurations.
Abstract: We investigate the problem of constructing a dynamics on edge-spin configurations which realizes a coupling between a Glauber dynamics of the Ising model and a dynamical evolution of the percolation configurations. We dream of constructing a Markov process on edge-spin configurations which is reversible with respect to the Ising-FK coupling measure, and such that the marginal on the spins is a Glauber dynamics, while the marginal on the edges is a Markovian evolution. We present two local dynamics, one which fulfills only the first condition and one which fulfills the first two conditions. We show next that our dream process is not feasible in general. We present a third dynamics, which is non local and fulfills the first and the third conditions. We finally present a localized version of this third dynamics, which can be seen as a contraction of the first dynamics.
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TL;DR: In this paper, the authors studied the exit time of a self-stabilizing McKean-Vlasov diffusion and showed that the first time that this diffusion leaves an interval of the form (d; +∞) (d verifying some assumptions) satisfies a Kramers' type law.
Abstract: The present paper is devoted to the study of a McKean-Vlasov diffusion oftype self-stabilizing. We obtain this model by taking the hydrodynamical limit of a mean-field system of particles. The main question that we study is the exit-time. We take a confining potential with two wells : a < 0 and b > 0. We start with a deterministic condition x0 > 0 and we show that the first time that this diffusion leaves an interval of the form (d; +∞) (d verifying some assumptions) satisfies a Kramers’type law. In other words, this time is exponentially equivalent to exp { 2 σ2H } as the diffusion coefficient σ goes to 0, H being the exit cost. Incidentally, we also prove that the solution of the granular media equation is trapped (for the 2-Wasserstein distance) in a ball centered around δb during a time at least exponentially equivalent to exp { 2 σ2H } .