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On fixed points of a generalized multidimensional affine recursion
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  • Published: 06 July 2012

On fixed points of a generalized multidimensional affine recursion

  • Mariusz Mirek1 

Probability Theory and Related Fields volume 156, pages 665–705 (2013)Cite this article

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Abstract

Let G be a multiplicative subsemigroup of the general linear group Gl \({(\mathbb{R}^d)}\) which consists of matrices with positive entries such that every column and every row contains a strictly positive element. Given a G-valued random matrix A, we consider the following generalized multidimensional affine equation

$$R\stackrel{\mathcal{D}}{=} \sum_{i=1}^N A_iR_i+B,$$

where N ≥ 2 is a fixed natural number, A 1, . . . , A N are independent copies of \({A, B \in \mathbb{R}^d}\) is a random vector with positive entries, and R 1, . . . , R N are independent copies of \({R \in \mathbb{R}^d}\) , which have also positive entries. Moreover, all of them are mutually independent and \({\stackrel{\mathcal{D}}{=}}\) stands for the equality in distribution. We will show with the aid of spectral theory developed by Guivarc’h and Le Page (Simplicité de spectres de Lyapounov et propriété d’isolation spectrale pour une famille d’opérateurs de transfert sur l’espace projectif. Random Walks and Geometry, Walter de Gruyter GmbH & Co. KG, Berlin, 2004; On matricial renewal theorems and tails of stationary measures for affine stochastic recursions, Preprint, 2011) and Kesten’s renewal theorem (Kesten in Ann Probab 2:355–386, 1974), that under appropriate conditions, there exists χ >  0 such that \({{\mathbb{P}(\{\langle R, u \rangle > t\})\asymp t^{-\chi}}}\) , as t → ∞, for every unit vector \({u \in \mathbb{S}^{d-1}}\) with positive entries.

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Acknowledgments

The results of this paper are part of the author’s PhD thesis, written under the supervision of Ewa Damek at the University of Wroclaw. I wish to thank her for many stimulating conversations and several helpful suggestions during the preparation of this paper. I would like also to thank Dariusz Buraczewski for beneficial discussions and comments. The author is grateful to the referee for a very careful reading of the manuscript and useful remarks that lead to the improvement of the presentation.

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This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

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Authors and Affiliations

  1. Institute of Mathematics, University of Wroclaw, 50-384, Wroclaw, Poland

    Mariusz Mirek

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  1. Mariusz Mirek
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Correspondence to Mariusz Mirek.

Additional information

This research project was partially supported by MNiSW Grant N N201 392337.

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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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Mirek, M. On fixed points of a generalized multidimensional affine recursion. Probab. Theory Relat. Fields 156, 665–705 (2013). https://doi.org/10.1007/s00440-012-0439-y

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  • Received: 23 October 2011

  • Revised: 26 May 2012

  • Published: 06 July 2012

  • Issue Date: August 2013

  • DOI: https://doi.org/10.1007/s00440-012-0439-y

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Keywords

  • Heavy tailed random variables
  • Renewal theory
  • Stationary measures
  • Markov chains
  • Spectral theory

Mathematics Subject Classification

  • 60H25
  • 60K05
  • 60J80
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