Tail-homogeneity of stationary measures for some multidimensional stochastic recursions

Abstract

We consider a stochastic recursion X n+1 = M n+1 X n + Q n+1, (\({n\in \mathbb {N}}\)), where (Q n , M n ) are i.i.d. random variables such that Q n are translations, M n are similarities of the Euclidean space \({\mathbb {R}^d}\) and \({X_n\in \mathbb {R}^d}\). In the present paper we show that if the recursion has a unique stationary measure ν with unbounded support then the weak limit of properly dilated ν exists and defines a homogeneous tail measure Λ. The structure of Λ is studied and the supports of ν and Λ are compared. In particular, we obtain a product formula for Λ.