Recurrence of multidimensional affine recursions in the critical case

Abstract

We prove, under different natural hypotheses, that the random multidimensional affine recursion $X_n=A_n{X}_{n-1}+B_n\in {\mathbb{R}}^d,n\ge 1$, 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".

Revised version, accepted for publication in Journal of Theoretical Probability, 28 pages