arXiv subjects

Olivier Durieu

Publications and source records attributed to Olivier Durieu.

At least 19 recordsLinked to original sources

arXiv2026

Limit distribution of the sample volume fraction of Boolean set

We study the limit distribution of the volume fraction estimator $\widehat p_{λ, A}$ (= the Lebesgue measure of the intersection $\mathcal{X}\cap (λA)$ of a random set $\mathcal{X}$ with a large observation set $λA$, divided by the Lebesgue measure of $λA$), as $λ\to \infty$, for a Boolean set $\mathcal{X}$ formed by uniformly scattered random grains $Ξ\subset \mathbb{R}^ν$. We obtain general conditions on generic grain set $Ξ$ under which $\widehat p_{λ, A}$ has an $α$-stable limit distribution with index $1 < α\le 2$. A large class of Boolean models with randomly homothetic grains satisfying these conditions is introduced. We also discuss the limit distribution of the sample volume fraction of a Boolean set observed on a large subset of a $ν_0$-dimensional $(1 \le ν_0 \le ν-1$) hyperplane of $\mathbb{R}^ν$.

math.PR↗

arXiv2024

Disjoint finite geodesics in first-passage percolation

We investigate first-passage percolation on the lattice $\Z^d$ for dimensions $d \geq 2$. Each edge $e$ of the graph is assigned an independent copy of a non-negative random variable $τ$. We only assume $¶[τ=0]0$ is explicit) for the probability of having two disjoint geodesics between two pairs of neighbouring vertices at distance $n$. Additionally, under more specific assumptions on the distribution of $τ$, we obtain similar lower bounds for the probability of having two disjoint geodesics (except for their starting and ending points) between the same two vertices.

math.PR↗

arXiv2021

Phase transition for extremes of a stochastic model with long-range dependence and multiplicative noise

We consider a stochastic process with long-range dependence perturbed by multiplicative noise. The marginal distributions of both the original process and the noise have regularly-varying tails, with tail indices $α,α'>0$, respectively. The original process is taken as the regularly-varying Karlin model, a recently investigated model that has long-range dependence characterized by a memory parameter $β\in(0,1)$. We establish limit theorems for the extremes of the model, and reveal a phase transition. In terms of the limit there are three different regimes: signal-dominance regime $α<α'β$, noise-dominance regime $α>α'β$, and critical regime $α= α'β$. As for the proof, we actually establish the same phase-transition phenomena for the so-called Poisson--Karlin model with multiplicative noise defined on generic metric spaces, and apply a Poissonization method to establish the limit theorems for the one-dimensional case as a consequence.

math.PR↗

arXiv2019

From infinite urn schemes to self-similar stable processes

We investigate the randomized Karlin model with parameter $β\in(0,1)$, which is based on an infinite urn scheme. It has been shown before that when the randomization is bounded, the so-called odd-occupancy process scales to a fractional Brownian motion with Hurst index $β/2\in(0,1/2)$. We show here that when the randomization is heavy-tailed with index $α\in(0,2)$, then the odd-occupancy process scales to a $(β/α)$-self-similar symmetric $α$-stable process with stationary increments.

math.PR↗

arXiv2018

A family of random sup-measures with long-range dependence

A family of self-similar and translation-invariant random sup-measures with long-range dependence are investigated. They are shown to arise as the limit of the empirical random sup-measure of a stationary heavy-tailed process, inspired by an infinite urn scheme, where same values are repeated at several random locations. The random sup-measure reflects the long-range dependence nature of the original process, and in particular characterizes how locations of extremes appear as long-range clusters represented by random closed sets. A limit theorem for the corresponding point-process convergence is established.

math.PR↗

arXiv2018

Generalized operator-scaling random ball model

This article introduces the operator-scaling random ball model, generalizing the isotropic random ball models investigated recently in the literature to anisotropic setup. The model is introduced as a generalized random field and results on weak convergence are established in the space of tempered distributions.

math.PR↗

arXiv2018

From random partitions to fractional Brownian sheets

We propose discrete random-field models that are based on random partitions of $\mathbb{N}^2$. The covariance structure of each random field is determined by the underlying random partition. Functional central limit theorems are established for the proposed models, and fractional Brownian sheets, with full range of Hurst indices, arise in the limit. Our models could be viewed as discrete analogues of fractional Brownian sheets, in the same spirit that the simple random walk is the discrete analogue of the Brownian motion.

math.PR↗

arXiv2017

Generalized random fields and Lévy's continuity theorem on the space of tempered distributions

In this note, we recall main properties of generalized random fields and present a proof of the continuity theorem of Paul Lévy for generalized random fields in the space of tempered distributions. This theorem was first proved by Fernique (1968) in a more general setting. The aim of this note is to provide a self-contained proof that in particular avoids the abstract theory of nuclear spaces.

math.PR↗

arXiv2015

From infinite urn schemes to decompositions of self-similar Gaussian processes

We investigate a special case of infinite urn schemes first considered by Karlin (1967), especially its occupancy and odd-occupancy processes. We first propose a natural randomization of these two processes and their decompositions. We then establish functional central limit theorems, showing that each randomized process and its components converge jointly to a decomposition of certain self-similar Gaussian process. In particular, the randomized occupancy process and its components converge jointly to the decomposition of a time-changed Brownian motion $\mathbb B(t^α), α\in(0,1)$, and the randomized odd-occupancy process and its components converge jointly to a decomposition of fractional Brownian motion with Hurst index $H\in(0,1/2)$. The decomposition in the latter case is a special case of the decompositions of bi-fractional Brownian motions recently investigated by Lei and Nualart (2009). The randomized odd-occupancy process can also be viewed as correlated random walks, and in particular as a complement to the model recently introduced by Hammond and Sheffield (2013) as discrete analogues of fractional Brownian motions.

math.PR↗

arXiv2015

Invariance principles for operator-scaling Gaussian random fields

Recently, Hammond and Sheffield introduced a model of correlated random walks that scale to fractional Brownian motions with long-range dependence. In this paper, we consider a natural generalization of this model to dimension $d\geq 2$. We define a $\mathbb Z^d$-indexed random field with dependence relations governed by an underlying random graph with vertices $\mathbb Z^d$, and we study the scaling limits of the partial sums of the random field over rectangular sets. An interesting phenomenon appears: depending on how fast the rectangular sets increase along different directions, different random fields arise in the limit. In particular, there is a critical regime where the limit random field is operator-scaling and inherits the full dependence structure of the discrete model, whereas in other regimes the limit random fields have at least one direction that has either invariant or independent increments, no longer reflecting the dependence structure in the discrete model. The limit random fields form a general class of operator-scaling Gaussian random fields. Their increments and path properties are investigated.

math.PR↗

arXiv2014

A Sequential Empirical Central Limit Theorem for Multiple Mixing Processes with Application to B-Geometrically Ergodic Markov Chains

We investigate the convergence in distribution of sequential empirical processes of dependent data indexed by a class of functions F. Our technique is suitable for processes that satisfy a multiple mixing condition on a space of functions which differs from the class F. This situation occurs in the case of data arising from dynamical systems or Markov chains, for which the Perron--Frobenius or Markov operator, respectively, has a spectral gap on a restricted space. We provide applications to iterative Lipschitz models that contract on average.

math.PR↗

arXiv2014

Approximating class approach for empirical processes of dependent sequences indexed by functions

We study weak convergence of empirical processes of dependent data $(X_i)_{i\geq0}$, indexed by classes of functions. Our results are especially suitable for data arising from dynamical systems and Markov chains, where the central limit theorem for partial sums of observables is commonly derived via the spectral gap technique. We are specifically interested in situations where the index class ${\mathcal{F}}$ is different from the class of functions $f$ for which we have good properties of the observables $(f(X_i))_{i\geq0}$. We introduce a new bracketing number to measure the size of the index class ${\mathcal{F}}$ which fits this setting. Our results apply to the empirical process of data $(X_i)_{i\geq0}$ satisfying a multiple mixing condition. This includes dynamical systems and Markov chains, if the Perron-Frobenius operator or the Markov operator has a spectral gap, but also extends beyond this class, for example, to ergodic torus automorphisms.

math.PR↗

arXiv2012

An Empirical Process Central Limit Theorem for Multidimensional Dependent Data

Let $(U_n(t))_{t\in\R^d}$ be the empirical process associated to an $\R^d$-valued stationary process $(X_i)_{i\ge 0}$. We give general conditions, which only involve processes $(f(X_i))_{i\ge 0}$ for a restricted class of functions $f$, under which weak convergence of $(U_n(t))_{t\in\R^d}$ can be proved. This is particularly useful when dealing with data arising from dynamical systems or functional of Markov chains. This result improves those of [DDV09] and [DD11], where the technique was first introduced, and provides new applications.

math.PR↗

arXiv2012

Empirical processes of iterated maps that contract on average

We consider a Markov chain obtained by random iterations of Lipschitz maps $T_i$ chosen with a probability $p_i(x)$ depending on the current position $x$. We assume this system has a property of "contraction on average", that is $\sum_i d(T_ix,T_iy)p_i(x) < ρd(x,y)$ for some $ρ<1$. In the present note, we study the weak convergence of the empirical process associated to this Markov chain.

math.PR↗

arXiv2011

Empirical Processes of Multidimensional Systems with Multiple Mixing Properties

We establish a multivariate empirical process central limit theorem for stationary $\R^d$-valued stochastic processes $(X_i)_{i\geq 1}$ under very weak conditions concerning the dependence structure of the process. As an application we can prove the empirical process CLT for ergodic torus automorphisms. Our results also apply to Markov chains and dynamical systems having a spectral gap on some Banach space of functions. Our proof uses a multivariate extension of the techniques introduced by Dehling, Durieu and Volný \cite{DehDurVol09} in the univariate case. As an important technical ingredient, we prove a $(2p)$th moment bound for partial sums in multiple mixing systems.

math.PR↗

arXiv2009

An Indicator Function Limit Theorem in Dynamical Systems

We show by a constructive proof that in all aperiodic dynamical system, for all sequences $(a_n)_{n\in\N}\subset\R_+$ such that $a_n\nearrow\infty$ and $\frac{a_n}{n}\to 0$ as $n\to\infty$, there exists a set $A\in\A$ having the property that the sequence of the distributions of $(\frac{1}{a_{n}}S_{n}(\ind_A-μ(A)))_{n\in\N}$ is dense in the space of all probability measures on $\R$.

math.DS↗

arXiv2009

On Sums of Indicator Functions in Dynamical Systems

In this paper, we are interested in the limit theorem question for sums of indicator functions. We show that in every aperiodic dynamical system, for every increasing sequence $(a_n)_{n\in\N}\subset\R_+$ such that $a_n\nearrow\infty$ and $\frac{a_n}{n}\to 0$ as $n\to\infty$, there exists a measurable set $A$ such that the sequence of the distributions of the partial sums $\frac{1}{a_n}\sum_{i=0}^{n-1}(\ind_A-μ(A))\circ T^i$ is dense in the set of the probability measures on $\R$. Further, in the ergodic case, we prove that there exists a dense $G_δ$ of such sets.

math.DS↗

arXiv2008

A fourth moment inequality for functionals of stationary processes

In this paper, a fourth moment bound for partial sums of functional of strongly ergodic Markov chain is established. This type of inequality plays an important role in the study of empirical process invariance principle. This one is specially adapted to the technique of Dehling, Durieu and Volný (2008). The same moment bound can be proved for dynamical system whose transfer operator has some spectral properties. Examples of applications are given.

math.PR↗