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Tail homogeneity of invariant measures of multidimensional stochastic recursions in a critical case
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  • Published: 12 June 2012

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

  • Konrad Kolesko1 

Probability Theory and Related Fields volume 156, pages 593–612 (2013)Cite this article

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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 \({Q_n, X_n \in \mathbb{R}^d }\), 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.

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Acknowledgments

The results of this paper are part of the author’s PhD thesis, written under the supervision of Dariusz Buraczewski at the University of Wroclaw. I wish to thank him for many stimulating conversation, suggestions and corrections incorporated in this paper. Part of this paper was written when I was staying at the Graz University of Technology at the invitation of Wolfgang Woess whom I would like to thank for help and hospitality. I would also like to thank anonymous referee for valuable comments that allowed to shortened the proof and write it more clearly.

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Authors and Affiliations

  1. Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384, Wrocław, Poland

    Konrad Kolesko

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  1. Konrad Kolesko
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Correspondence to Konrad Kolesko.

Additional information

This research project has been partially supported by MNiSW grant N N201 610740 and also by FWF grant FWF-P19115-N18.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Kolesko, K. Tail homogeneity of invariant measures of multidimensional stochastic recursions in a critical case. Probab. Theory Relat. Fields 156, 593–612 (2013). https://doi.org/10.1007/s00440-012-0437-0

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  • Received: 19 May 2011

  • Revised: 14 March 2012

  • Published: 12 June 2012

  • Issue Date: August 2013

  • DOI: https://doi.org/10.1007/s00440-012-0437-0

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Keywords

  • Random walk
  • Affine group
  • Tail homogeneity
  • Invariant measure

Mathematics Subject Classification (2000)

  • 60B15
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