Probability Theory and Related Fields

, Volume 156, Issue 3, pp 593-612

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

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

  • Konrad KoleskoAffiliated withInstytut Matematyczny, Uniwersytet Wrocławski Email author 


We 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.


Random walk Affine group Tail homogeneity Invariant measure

Mathematics Subject Classification (2000)