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Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps
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  • Published: 21 July 2010

Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps

  • Mariusz Mirek1 

Probability Theory and Related Fields volume 151, pages 705–734 (2011)Cite this article

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Abstract

We consider the Markov chain \({\{X_n^x\}_{n=0}^\infty}\) on \({\mathbb{R}^d}\) defined by the stochastic recursion \({X_{n}^{x}= \psi_{\theta_{n}} (X_{n-1}^{x})}\), starting at \({x\in\mathbb{R}^d}\), where θ 1, θ 2, . . . are i.i.d. random variables taking their values in a metric space \({(\Theta, \mathfrak{r})}\), and \({\psi_{\theta_{n}} :\mathbb{R}^d\mapsto\mathbb{R}^d}\) are Lipschitz maps. Assume that the Markov chain has a unique stationary measure ν. Under appropriate assumptions on \({\psi_{\theta_n}}\), we will show that the measure ν has a heavy tail with the exponent α > 0 i.e. \({\nu(\{x\in\mathbb{R}^d: |x| > t\})\asymp t^{-\alpha}}\). Using this result we show that properly normalized Birkhoff sums \({S_n^x=\sum_{k=1}^n X_k^x}\), converge in law to an α-stable law for \({\alpha\in(0, 2]}\).

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Authors and Affiliations

  1. Institute of Mathematics, University of Wroclaw, Plac Grunwaldzki 2/4, 50-384, Wroclaw, Poland

    Mariusz Mirek

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  1. Mariusz Mirek
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Corresponding author

Correspondence to Mariusz Mirek.

Additional information

In memory of Andrzej Hulanicki.

This research project has been partially supported by Marie Curie Transfer of Knowledge Fellowship Harmonic Analysis, Nonlinear Analysis and Probability (contract number MTKD-CT-2004-013389) and by KBN grant N201 012 31/1020.

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Mirek, M. Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps. Probab. Theory Relat. Fields 151, 705–734 (2011). https://doi.org/10.1007/s00440-010-0312-9

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  • Received: 15 October 2009

  • Revised: 29 June 2010

  • Published: 21 July 2010

  • Issue Date: December 2011

  • DOI: https://doi.org/10.1007/s00440-010-0312-9

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Mathematics Subject Classification (2000)

  • 60J10
  • 60K05
  • 60F05
  • 60B15
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