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Tail-homogeneity of stationary measures for some multidimensional stochastic recursions
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  • Published: 20 August 2008

Tail-homogeneity of stationary measures for some multidimensional stochastic recursions

  • Dariusz Buraczewski1,
  • Ewa Damek1,
  • Yves Guivarc’h2,
  • Andrzej Hulanicki1 &
  • …
  • Roman Urban1 

Probability Theory and Related Fields volume 145, pages 385–420 (2009)Cite this article

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Abstract

We consider a stochastic recursion X n+1 = M n+1 X n  + Q n+1, (\({n\in \mathbb {N}}\)), where (Q n , M n ) are i.i.d. random variables such that Q n are translations, M n are similarities of the Euclidean space \({\mathbb {R}^d}\) and \({X_n\in \mathbb {R}^d}\). In the present paper we show that if the recursion has a unique stationary measure ν with unbounded support then the weak limit of properly dilated ν exists and defines a homogeneous tail measure Λ. The structure of Λ is studied and the supports of ν and Λ are compared. In particular, we obtain a product formula for Λ.

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Authors and Affiliations

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

    Dariusz Buraczewski, Ewa Damek, Andrzej Hulanicki & Roman Urban

  2. IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042, Rennes Cedex, France

    Yves Guivarc’h

Authors
  1. Dariusz Buraczewski
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  2. Ewa Damek
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  3. Yves Guivarc’h
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  4. Andrzej Hulanicki
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  5. Roman Urban
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Corresponding author

Correspondence to Dariusz Buraczewski.

Additional information

This research project has been partially supported by European Commission via IHP Network 2002–2006 Harmonic Analysis and Related Problems(contract Number: HPRN-CT-2001-00273-HARP) and a Marie Curie Transfer of Knowledge Fellowship Harmonic Analysis, Nonlinear Analysis and Probability (contract number MTKD-CT-2004-013389). D. Buraczewski, E. Damek, A. Hulanicki and R. Urban were also supported by KBN grants 1 P03A 018 26 and N201 012 31/1020.

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Buraczewski, D., Damek, E., Guivarc’h, Y. et al. Tail-homogeneity of stationary measures for some multidimensional stochastic recursions. Probab. Theory Relat. Fields 145, 385–420 (2009). https://doi.org/10.1007/s00440-008-0172-8

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  • Received: 24 July 2007

  • Revised: 10 July 2008

  • Published: 20 August 2008

  • Issue Date: November 2009

  • DOI: https://doi.org/10.1007/s00440-008-0172-8

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Mathematics Subject Classification (2000)

  • 60J10
  • 60K05
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