On the affine recursion on ${\mathbb{R}}_{+}^d$

Abstract

We fix $d\ge 2$ and denote $\mathcal{S}$ the semi-group of $d\times d$ matrices with non negative entries. We consider a sequence $(A_n,B_n{)}_{n\ge 1}$ of i. i. d. random variables with values in $\mathcal{S}\times {\mathbb{R}}_{+}^d$ and study the asymptotic behavior of the Markov chain $(X_n{)}_{n\ge 0}$ 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 $\lambda$ on $({\mathbb{R}}^{+}{)}^d$ which is invariant for the chain $(X_n{)}_{n\ge 0}$. The existence of $\lambda$ 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 .