On invariant measures of stochastic recursions in a critical case

Abstract

We consider an autoregressive model on 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 ℝ×ℝ⁺ and $\mathbb{E}[\log A_{1}]=0$ (critical case). It was proved by Babillot, Bougerol and Elie that there exists a unique invariant Radon measure of the process {X_n}. The aim of the paper is to investigate its behavior at infinity. We describe also stationary measures of two other stochastic recursions, including one arising in queuing theory.