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

We consider the autoregressive model on $\R^d$ defined by the following stochastic recursion $X_n = A_n X_{n-1}+B_n$, where $\{(B_n,A_n)\}$ are i.i.d. random variables valued in $\R^d\times \R^+$. The critical case, when $\E\big[\log…

We consider an autoregressive model on $\mathbb{R}$ defined by the recurrence equation $X_n=A_nX_{n-1}+B_n$, where $\{(B_n,A_n)\}$ are i.i.d. random variables valued in $\mathbb{R}\times\mathbb{R}^+$ and $\mathbb {E}[\log A_1]=0$ (critical…

We consider stochastic dynamical systems on ${\mathbb{R}}$, that is, random processes defined by $X_n^x=\Psi_n(X_{n-1}^x)$, $X_0^x=x$, where $\Psi _n$ are i.i.d. random continuous transformations of some unbounded closed subset of…

We consider a Markov chain $\{X_n\}_{n=0}^\8$ on $\R^d$ defined by the stochastic recursion $X_{n}=M_n X_{n-1}+Q_n$, where $(Q_n,M_n)$ are i.i.d. random variables taking values in the affine group $H=\R^d\rtimes {\rm GL}(\R^d)$. Assume that…

Let $\Phi_n$ be an i.i.d. sequence of Lipschitz mappings of $\R^d$. We study the Markov chain $\{X_n^x\}_{n=0}^\infty$ on $\R^d$ defined by the recursion $X_n^x = \Phi_n(X^x_{n-1})$, $n\in\N$, $X_0^x=x\in\R^d$. We assume that…

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…

Let ${\bf M}=(M_1,\ldots, M_k)$ be a tuple of real $d\times d$ matrices. Under certain irreducibility assumptions, we give checkable criteria for deciding whether ${\bf M}$ possesses the following property: there exist two constants…

We consider random walks on the group of orientation-preserving homeomorphisms of the real line ${\mathbb R}$. In particular, the fundamental question of uniqueness of an invariant measure of the generated process is raised. This problem…

Let $G$ be a multiplicative subsemigroup of the general linear group $\Gl(\mathbb{R}^d)$ which consists of matrices with positive entries such that every column and every row contains a strictly positive element. Given a $G$--valued random…

Consider a proper metric space X and a sequence of i.i.d. random continuous mappings F_n from X to X. It induces the stochastic dynamical system (SDS) X_n^x = F_n(X_{n-1}^x) starting at x in X. In this paper, we study existence and…

This paper is about the rate of convergence of the Markov chain $X_{n+1}=AX_{n}+B_{n}$ (mod $p$), where $A$ is an integer matrix with nonzero eigenvalues and ${B_{n}}_{n}$ is a sequence of independent and identically distributed integer…

Given a sequence $(M_{k}, Q_{k})_{k\ge 1}$ of independent, identically distributed ran\-dom vectors with nonnegative components, we consider the recursive Markov chain $(X_{n})_{n\ge 0}$, defined by the random difference equation…

We study the asymptotic behaviour of Markov chains $(X_n,\eta_n)$ on $\mathbb{Z}_+ \times S$, where $\mathbb{Z}_+$ is the non-negative integers and $S$ is a finite set. Neither coordinate is assumed to be Markov. We assume a moments bound…

Consider continuous-time linear switched systems on R^n associated with compact convex sets of matrices. When the system is irreducible and the largest Lyapunov exponent is equal to zero, there always exists a Barabanov norm (i.e. a norm…

Given an i.i.d. sequence $\{A_n(\omega)\}_{n\ge 1}$ of invertible matrices and a random matrix $B(\omega)$, we consider the random matrix sequences inductively defined by $S_n(\omega) = A_n(\omega)S_{n-1}(\omega)$ and $T_n(\omega) =…

Let $X_1,X_2, \ldots $ be a sequence of $i.i.d$ real (complex) $d \times d $ invertible random matrices with common distribution $\mu$ and $\sigma_1(n), \sigma_2(n), \ldots , \sigma_d(n)$ be the singular values, $\lambda_1(n), \lambda_2(n),…

Given a sequence of i.i.d. random functions $\Psi_{n}:\mathbb{R}\to\mathbb{R}$, $n\in\mathbb{N}$, we consider the iterated function system and Markov chain which is recursively defined by $X_{0}^{x}:=x$ and…

Let $(\Omega,\mathcal{F}, \mathbb{P})$ be a probability space and $E$ be a finite set. Assume that $X=(X_n)$ is an irreducible and aperiodic Markov chain, defined on $(\Omega,\mathcal{F}, \mathbb{P})$, with values in $E$ and with transition…

We consider a certain infinite product of random $2 \times 2$ matrices appearing in the solution of some $1$ and $1+1$ dimensional disordered models in statistical mechanics, which depends on a parameter $\varepsilon>0$ and on a real random…

We present a recurrence-transience classification for discrete-time Markov chains on manifolds with negative curvature. Our classification depends only on geometric quantities associated to the increments of the chain, defined via the…