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

Abstract

We consider the stochastic recursion X n+1 = M n+1 X n + Q n+1 , (n ∈ N), where Q n , X n ∈ R d , M n are similarities of the Euclidean space 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 E[log |M|] = 0 i.e. in the so called critical case.