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Homogeneity at Infinity of Stationary Solutions of Multivariate Affine Stochastic Recursions

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Random Matrices and Iterated Random Functions

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 53))

Abstract

We consider a d-dimensional affine stochastic recursion of general type corresponding to the relation

$$\displaystyle{ X_{n+1} = A_{n+1}X_{n} + B_{n+1},\quad X_{0} = x. }$$
(S)

Under natural conditions, this recursion has a unique stationary solution R, which is unbounded. If d > 2, we sketch a proof of the fact that R belongs to the domain of attraction of a stable law which depends essentially of the linear part of the recursion. The proof is based on renewal theorems for products of random matrices, radial Fourier analysis in the vector space \({\mathbb{R}}^{d}\), and spectral gap properties for convolution operators on the corresponding projective space. We state the corresponding simpler result for d = 1.

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Authors and Affiliations

  1. Institut de Recherche Mathématiques de Rennes (IRMAR), Université de Rennes 1, UFR de mathématiques, 263 Avenue du General Leclerc CS 74205, 35042, Rennes, France

    Yves Guivarc’h

  2. Université de Bretagne Sud, LMBA, Campus de Tohannic, 573, 56017, Vannes, France

    Émile Le Page

Authors
  1. Yves Guivarc’h
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  2. Émile Le Page
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Correspondence to Yves Guivarc’h .

Editor information

Editors and Affiliations

  1. Westphalian Wilhelms University Münster Institute of Mathematics and Informatics, Münster, Germany

    Gerold Alsmeyer

  2. Westphalian Wilhelms University Münster Institute for Mathematical Statistics, Münster, Germany

    Matthias Löwe

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Guivarc’h, Y., Page, É.L. (2013). Homogeneity at Infinity of Stationary Solutions of Multivariate Affine Stochastic Recursions. In: Alsmeyer, G., Löwe, M. (eds) Random Matrices and Iterated Random Functions. Springer Proceedings in Mathematics & Statistics, vol 53. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38806-4_6

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