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Marc Peigné

Publications and source records attributed to Marc Peigné.

At least 19 recordsLinked to original sources
arXiv2026

Recurrence of multidimensional affine recursions in the critical case

We prove, under different natural hypotheses, that the random multidimensional affine recursion $X_n=A_nX_{n-1}+B_n\in\mathbb{R}^d, n \geq 1$, is recurrent in the critical case. In particular we cover the cases where the matrices $A_n$ are similarities, invertible, rank 1 or with non negative coefficients. These results are a consequence of a criterion of recurrence for a large class of affine recursions on $\mathbb R^d$, based on some moment assumptions of the so-called ``reverse norm control random variable".

math.PR↗
arXiv2026

A local limit theorem for nonlattice multidimensional random walks in cones

We study the asymptotic behavior of a nonlattice random walk in a general cone of $R^d$ . Following the approach initiated by D. Denisov and V. Wachtel in [8], we use a strong approximation of random walks by the Brownian motion and prove local limit theorems, combining integral theorems for random walks in cones with classical theorems for unrestricted random walks.

math.PR↗
arXiv2024

On the rate of convergence in the weak invariance principle for dependent random variables with applications to Markov chains

We prove an invariance principle for non-stationary random processes and establish a rate of convergence under a new type of mixing condition. The dependence is exponentially decaying in the gap between the past and the future and is controlled by an assumption on the characteristic function of the finite dimensional increments of the process. The distinct feature of the new mixing condition is that the dependence increases exponentially in the dimension of the increments. The proposed mixing property is particularly suited for processes whose behavior can be described in terms of spectral properties of some related family of operators. Several examples are discussed. We also work out explicit expressions for the constants involved in the bounds. When applied to Markov chains our result specifies the dependence of the constants on the properties of the underlying Banach space and on the initial state of the chain.

math.PR↗
arXiv2023

A functional limit theorem for lattice oscillating random walk

The paper is devoted to an invariance principle for Kemperman's model of oscillating random walk on $\mathbb{Z}$. This result appears as an extension of the invariance principal theorem for classical random walks on $\mathbb{Z}$ or reflected random walks on $\mathbb{N}_0$. Relying on some natural Markov sub-process which takes into account the oscillation of the random walks between $\mathbb{Z}^-$ and $\mathbb{Z}^+$, we first construct an aperiodic sequence of renewal operators acting on a suitable Banach space and then apply a powerful theorem proved by S. Gouëzel.

math.PR↗
arXiv2023

Exotic local limit theorems at the phase transition in free products

We construct random walks on free products of the form Z 3 * Z d , with d = 5 or 6 which are divergent and not spectrally positive recurrent. We then derive a local limit theorem for these random walks, proving that $μ$ * n (e) $\sim$ CR --n n --5/3 if d = 5 and $μ$ * n (e) $\sim$ CR --n n --3/2 log(n) --1/2 if d = 6, where $μ$ * n is the nth convolution power of $μ$ and R is the inverse of the spectral radius of $μ$. This disproves a result of Candellero and Gilch [7] and a result of the authors of this paper that was stated in a rst version of [11]. This also shows that the classication of local limit theorems on free products of the form Z d 1 * Z d 2 or more generally on relatively hyperbolic groups with respect to virtually abelian subgroups is incomplete.

math.DS↗
arXiv2023

A local limit theorem for convergent random walks on relatively hyperbolic groups

We study random walks on relatively hyperbolic groups whose law is convergent, in the sense that the derivative of its Green function is finite at the spectral radius.When parabolic subgroups are virtually abelian, we prove that for such a random walk satisfies a local limit theorem of the form $p_n(e, e)\sim CR^{-n}n^{-d/2}$, where $p_n(e, e)$ is the probability of returning to the origin at time $n$, $R$ is the inverse of the spectral radius of the random walk and $d$ is the minimal rank of a parabolic subgroup along which the random walk is spectrally degenerate.This concludes the classification all possible behaviour for $p_n(e, e)$ on such groups.

math.DS↗
arXiv2021

Limit theorem for reflected random walks

Let $ξ$ n , n $\in$ N be a sequence of i.i.d. random variables with values in Z. The associated random walk on Z is S(n) = $ξ$ 1 + $\times$ $\times$ $\times$ + $ξ$ n+1 and the corresponding "reflected walk" on N 0 is the Markov chain X(n), n $\in$ N, given by X(0) = x $\in$ N 0 and X(n + 1) = |X(n) + $ξ$ n+1 | for n $\ge$ 0. It is well know that the reflected walk (X(n)) n$\ge$0 is null-recurrent when the $ξ$ n are square integrable and centered. In this paper, we prove that the process (X(n)) n$\ge$0 , properly rescaled, converges in distribution towards the reflected Brownian motion on R + , when E[$ξ$ 2 n ] < +$\infty$, E[(max(0, --$ξ$ n) 3 ] < +$\infty$ and the $ξ$ n are aperiodic and centered.

math.PR↗
arXiv2021

Recurrence of 2-dimensional queueing processes, and random walk exit times from the quadrant

Let $X = (X_1, X_2)$ be a 2-dimensional random variable and $X(n), n \in \mathbb{N}$ a sequence of i.i.d. copies of $X$. The associated random walk is $S(n)= X(1) + \cdots +X(n)$. The corresponding absorbed-reflected walk $W(n), n \in \mathbb{N}$ in the first quadrant is given by $W(0) = x \in \mathbb{R}_+^2$ and $W(n) = \max \{ 0, W(n-1) - X(n) \}$, where the maximum is taken coordinate-wise. This is often called the Lindley process and models the waiting times in a two-server queue. We characterize recurrence of this process, assuming suitable, rather mild moment conditions on $X$. It turns out that this is directly related with the tail asymptotics of the exit time of the random walk $x + S(n)$ from the quadrant, so that the main part of this paper is devoted to an analysis of that exit time in relation with the drift vector, i.e., the expectation of $X$.

math.PR↗
arXiv2020

On the asymptotic behavior of the Diaconis and Freedman's chain in a multidimensional simplex

In this paper, we give out a setting of an Diaconis and Freedman's chain in a multidimensional simplex and consider its asymptotic behavior. By using techniques in random iterated functions theory and quasi-compact operators theory, we first give out some sufficient conditions which ensure the existence and uniqueness of an invariant probability measure. In some particular cases, we give out explicit formulas of the invariant probability density. Moreover, we completely classify all behaviors of this chain in dimensional two. Eventually, some other settings of the chain are discussed.

math.PR↗
arXiv2020

On the affine recursion on $\mathbb R_+^d$

We fix $d \geq 2$ and denote $\mathcal S$ the semi-group of $d \times d$ matrices with non negative entries. We consider a sequence $(A_n, B_n)_{n \geq 1} $ of i. i. d. random variables with values in $\mathcal S\times \mathbb R_+^d$ and study the asymptotic behavior of the Markov chain $(X_n)_{n \geq 0}$ on $ \mathbb R_+^d$ defined by: \[ \forall n \geq 0, \qquad X_{n+1}=A_{n+1}X_n+B_{n+1}, \] where $X_0$ is a fixed random variable. We assume that the Lyapunov exponent of the matrices $A_n$ equals $0$ and prove, under quite general hypotheses, that there exists a unique (infinite) Radon measure $λ$ on $(\mathbb R^+)^d$ which is invariant for the chain $(X_n)_{n \geq 0}$. The existence of $λ$ relies on a recent work by T.D.C. Pham about fluctuations of the norm of product of random matrices . Its unicity is a consequence of a general property, called "local contractivity", highlighted about 20 years ago by M. Babillot, Ph. Bougerol et L. Elie in the case of the one dimensional affine recursion .

math.PR↗
arXiv2017

Counting for some convergent groups

We present examples of geometrically finite manifolds with pinched negative curvature, whose geodesic flow has infinite non-ergodic Bowen-Margulis measure and whose Poincaré series converges at the critical exponent $δ_Γ$. We obtain an explicit asymptotic for their orbital growth function. Namely, for any $α\in ]1, 2[ $ and any slowly varying function $L : \mathbb R\to (0, +\infty)$, we construct $N$-dimensional Hadamard manifolds $(X, g)$ of negative and pinched curvature, whose group of oriented isometries admits convergent geometrically finite subgroups $Γ$ such that, as $R\to +\infty$, $$ N_Γ(R):= \#\left\{γ\in Γ\; ; \; d(o, γ\cdot o)\leq R\right\} \sim C_Γ\frac{L(R)}{R^α} \ e^{δ_ΓR}, $$ for some constant $C_Γ>0$.

math.DS↗
arXiv2017

Iterated function systems with place dependent probabilities and application to the Diaconis-Friedman's chain on [0,1]

We study Markov chains generated by iterated Lipschitz functions systems with possibly place dependent probabilities. Under general conditions, we prove uniqueness of the invariant probability measure for the associated Markov chain, by using quasi-compact linear operators technics. We use the same approach to describe the behavior of the Diaconis-Friedman's chain on [0,1] with possibly place dependent probabilities.

math.PR↗
arXiv2016

Conditioned limit theorems for products of random matrices

Consider the product $G_{n}=g_{n} ... g_{1}$ of the random matrices $g_{1},...,g_{n}$ in $GL(d,\mathbb{R}) $ and the random process $ G_{n}v=g_{n}... g_{1}v$ in $\mathbb{R}^{d}$ starting at point $v\in \mathbb{R}^{d}\smallsetminus \{0\} .$ It is well known that under appropriate assumptions, the sequence $(\log \Vert G_{n}v\Vert)_{n\geq 1}$ behaves like a sum of i.i.d.\ r.v.'s and satisfies standard classical properties such as the law of large numbers, law of iterated logarithm and the central limit theorem. Denote by $\mathbb{B}$ the closed unit ball in $\mathbb{R}^{d}$ and by $\mathbb{B}^{c}$ its complement. For any $v\in \mathbb{B}^{c}$ define the exit time of the random process $G_{n}v$ from $\mathbb{B}^{c}$ by $τ_{v}=\min \{n\geq 1:G_{n}v\in \mathbb{B}\} .$ We establish the asymptotic as $n \to \infty $ of the probability of the event $\{τ_{v}>n\} $ and find the limit law for the quantity $\frac{1}{\sqrt{n}} \log \Vert G_{n}v\Vert $ conditioned that $τ_{v}>n.$

math.PR↗
arXiv2015

Harmonic functions on multiplicative graphs and inverse Pitman transform on infinite random paths

We introduce and characterize central probability distributions on Littelmann paths. Next we establish a law of large numbers and a central limit theorem for the generalized Pitmann transform. We then study harmonic functions on multiplicative graphs defined from the tensor powers of finite-dimensional Lie algebras representations. Finally, we show there exists an inverse of the generalized Pitman transform defined almost surely on the set of infinite paths remaining in the Weyl chamber and explain how it can be computed.

math.CO↗
arXiv2013

Conditioned random walks from Kac-Moody root systems

Random paths are time continuous interpolations of random walks. By using Littelmann path model, we associate to each irreducible highest weight module of a Kac Moody algebra g a random path W. Under suitable hypotheses, we make explicit the probability of the event E: W never exits the Weyl chamber of g. We then give the law of the random walk defined by W conditioned by the event E and proves this law can be recovered by applying to W the generalized Pitmann transform introduced by Biane, Bougerol and O'Connell. This generalizes the main results of [10] and [16] to Kac Moody root systems and arbitrary highest weight modules. Moreover, we use here a completely new approach by exploiting the symmetry of our construction under the action of the Weyl group of g rather than renewal theory and Doob's theorem on Martin kernels.

math.CO↗
arXiv2013

Some aspects of fluctuations of random walks on R and applications to random walks on R+ with non-elastic reflection at 0

In this article we refine well-known results concerning the fluctuations of one-dimensional random walks. More precisely, if $(S_n)_{n \geq 0}$ is a random walk starting from 0 and $r\geq 0$, we obtain the precise asymptotic behavior as $n\to\infty$ of $\mathbb P[τ^{>r}=n, S_n\in K]$ and $\mathbb P[τ^{>r}>n, S_n\in K]$, where $τ^{>r}$ is the first time that the random walk reaches the set $]r,\infty[$, and $K$ is a compact set. Our assumptions on the jumps of the random walks are optimal. Our results give an answer to a question of Lalley stated in [9], and are applied to obtain the asymptotic behavior of the return probabilities for random walks on $\mathbb R^+$ with non-elastic reflection at 0.

math.PR↗
arXiv2012

Return Probabilities for the Reflected Random Walk on $\mathbb N_0$

Let $(Y_n)$ be a sequence of i.i.d. $\mathbb Z$-valued random variables with law $μ$. The reflected random walk $(X_n)$ is defined recursively by $X_0=x \in \mathbb N_0, X_{n+1}=|X_n+Y_{n+1}|$. Under mild hypotheses on the law $μ$, it is proved that, for any $ y \in \mathbb N_0$, as $n \to +\infty$, one gets $\mathbb P_x[X_n=y]\sim C_{x, y} R^{-n} n^{-3/2}$ when $\sum_{k\in \mathbb Z} kμ(k) >0$ and $\mathbb P_x[X_n=y]\sim C_{y} n^{-1/2}$ when $\sum_{k\in \mathbb Z} kμ(k) =0$, for some constants $R, C_{x, y}$ and $C_y >0$.

math.PR↗