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Convergence to stable laws for a class of multidimensional stochastic recursions
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  • Published: 10 June 2009

Convergence to stable laws for a class of multidimensional stochastic recursions

  • Dariusz Buraczewski1,
  • Ewa Damek1 &
  • Yves Guivarc’h2 

Probability Theory and Related Fields volume 148, pages 333–402 (2010)Cite this article

Abstract

We consider a Markov chain \({\{X_n\}_{n=0}^\infty}\) on \({\mathbb R^d}\) defined by the stochastic recursion X n  = M n X n-1 + Q n , where (Q n , M n ) are i.i.d. random variables taking values in the affine group \({A(\mathbb R^d)=\mathbb R^d\rtimes {\rm GL}(\mathbb R^d)}\). Assume that M n takes values in the group of similarities of \({\mathbb R^d}\), and the Markov chain has a unique stationary measure ν, which has unbounded support. We denote by |M n | the expansion coefficient of M n and we assume \({\mathbb E [|M|^\alpha]=1}\) for some positive α. We show that the partial sums \({S_n=\sum_{k=0}^n X_k}\), properly normalized, converge to a normal law (α ≥ 2) or to an infinitely divisible law, which is stable in a natural sense (α < 2). These laws are fully nondegenerate, if ν is not supported on an affine hyperplane. Under an aperiodicity hypothesis, we prove also a local limit theorem for the sums S n . If α ≤ 2, proofs are based on the homogeneity at infinity of ν and on a detailed spectral analysis of a family of Fourier operators P v considered as perturbations of the transition operator P of the chain {X n }. The characteristic function of the limit law has a simple expression in terms of moments of ν (α > 2) or of the tails of ν and of stationary measure for an associated Markov operator (α ≤ 2). We extend the results to the situation where M n is a random generalized similarity.

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Author information

Authors and Affiliations

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

    Dariusz Buraczewski & Ewa Damek

  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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Corresponding author

Correspondence to Dariusz Buraczewski.

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). D. Buraczewski and E. Damek were also supported by MNiSW grant N201 012 31/1020.

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Buraczewski, D., Damek, E. & Guivarc’h, Y. Convergence to stable laws for a class of multidimensional stochastic recursions. Probab. Theory Relat. Fields 148, 333–402 (2010). https://doi.org/10.1007/s00440-009-0233-7

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  • Received: 10 November 2008

  • Revised: 15 May 2009

  • Published: 10 June 2009

  • Issue Date: November 2010

  • DOI: https://doi.org/10.1007/s00440-009-0233-7

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Mathematics Subject Classification (2000)

  • 60F05
  • 60J05
  • 60B15
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