On the invariant measure of the random difference equation $X_n=A_n{X}_{n-1}+B_n$ in the critical case

Abstract

We consider the autoregressive model on ${\mathbb{R}}^d$ 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 ${\mathbb{R}}^d\times {\mathbb{R}}^{+}$. The critical case, when $\E\big[\log A_1\big]=0$, was studied by Babillot, Bougeorol and Elie, who proved that there exists a unique invariant Radon measure $\nu$ for the Markov chain $\{X_n\}$. In the present paper we prove that the weak limit of properly dilated measure $\nu$ exists and defines a homogeneous measure on ${\mathbb{R}}^d\setminus \{0\}$.