Tail homogeneity of invariant measures of multidimensional stochastic recursions in a critical case

Abstract

AbstractWe consider the stochastic recursion $${X_{n+1} = M_{n+1}X_{n} + Q_{n+1}, (n \in \mathbb{N})}$$, where $${Q_n, X_n \in \mathbb{R}^d }$$, M n are similarities of the Euclidean space $${ \mathbb{R}^d }$$ and (Q n , M n ) are i.i.d. We study asymptotic properties at infinity of the invariant measure for the Markov chain X n under assumption $${\mathbb{E}{[\log|M|]}=0}$$ i.e. in the so called critical case.