The random difference equation $X\sb n=A\sb nX\sb {n-1}+B\sb n$ in the critical case

Abstract

Let $(B_n, A_n)_{n \geq 1}$ be a sequence of i.i.d. random variables with values in $\mathbf{R}^d \times \mathbf{R}_*^+$. The Markov chain on $\mathbf{R}^d$ which satisfies the random equa tion $X_n = A_n X_{n-1} + B_n$ is studied when $\mathbf{E}(\log A_1) = 0$. No density assumption on the distribution of $(B_1, A_1)$ is made. The main results are recurrence of the Markov chain $X_n$, stability properties of the paths, existence and uniqueness of a Radon invariant measure and a limit theorem for the occupation times. The results rely on a renewal theorem for the process $(X_n, A_n \dots A_1)$.