Probability Theory and Related Fields

, Volume 156, Issue 3, pp 593–612

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

Open AccessArticle

DOI: 10.1007/s00440-012-0437-0

Cite this article as:
Kolesko, K. Probab. Theory Relat. Fields (2013) 156: 593. doi:10.1007/s00440-012-0437-0
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Abstract

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

Keywords

Random walkAffine groupTail homogeneityInvariant measure

Mathematics Subject Classification (2000)

60B15
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Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Instytut MatematycznyUniwersytet WrocławskiWrocławPoland