Comptes Rendus
Probability Theory
On the multidimensional stochastic equation Yn+1=AnYn+Bn
Comptes Rendus. Mathématique, Volume 339 (2004) no. 7, pp. 499-502.

We study the behavior at infinity of the tail of the stationary solution of a multidimensional linear auto-regressive process with random coefficients. We exhibit an extended class of multiplicative coefficients satisfying a condition of irreducibility and proximality that yield to a heavy tail behavior.

On étudie le comportement à l'infini de la queue de la solution stationnaire d'un processus auto-régressif linéaire multidimensionnel à coefficients aléatoires. On donne une vaste classe de coefficients multiplicatifs vérifiant une condition d'irréductibilité et de proximalité qui conduisent à un comportement de type queue polynomiale.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.07.024
Benoîte de Saporta 1; Yves Guivarc'h 1; Emile Le Page 2

1 IRMAR, université de Rennes I, campus de Beaulieu, 35042 Rennes cedex, France
2 LMAM, université de Bretagne Sud, centre Yves Coppens, campus de Tohannic, BP 573, 56017 Vannes, France
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Benoîte de Saporta; Yves Guivarc'h; Emile Le Page. On the multidimensional stochastic equation $ {Y}_{n+1}={A}_{n}{Y}_{n}+{B}_{n}$. Comptes Rendus. Mathématique, Volume 339 (2004) no. 7, pp. 499-502. doi : 10.1016/j.crma.2004.07.024. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.07.024/

[1] A. Brandt The stochastic equation Yn+1=AnYn+Bn with stationary coefficients, Adv. Appl. Probab., Volume 18 (1986), pp. 211-220

[2] H. Furstenberg Noncommuting random products, Trans. Amer. Math. Soc., Volume 108 (1963), pp. 377-428

[3] H. Furstenberg Boundary theory and stochastic processes on homogeneous spaces, Harmonic Analysis on Homogeneous Spaces, Proc. Sympos. Pure Math., vol. XXVI, American Mathematical Society, 1973, pp. 193-229

[4] C.M. Goldie Implicit renewal theory and tails of solutions of random equations, Ann. Appl. Probab., Volume 1 (1991), pp. 26-166

[5] I.Ya. Goldsheid; Y. Guivarc'h Zariski closure and the dimension of the Gaussian law of the product of random matrices, Probab. Theory Related Fields, Volume 105 (1996), pp. 109-142

[6] Y. Guivarc'h; E. Le Page Simplicité de spectres de Lyapunov et propriété d'isolation spectrale pour une famille d'opérateurs de transfert sur l'espace projectif (V. Kaimanovitch, ed.), Random Walks and Geometry, Workshop Vienna 2001, De Gruyter, 2004, pp. 181-259

[7] Y. Guivarc'h; A. Raugi Products of random matrices: convergence theorems, Random matrices and their applications, Contemp. Math., Volume 50 (1986), pp. 31-54

[8] H. Kesten Random difference equations and renewal theory for products of random matrices, Acta Math., Volume 131 (1973), pp. 207-248

[9] H. Kesten Renewal theory for functionals of a Markov chain with general state space, Ann. Probab., Volume 2 (1974), pp. 355-386

[10] E. Le Page, Théorèmes de renouvellement pour les produits de matrices aléatoires. Equations aux différences aléatoires, Séminaires de probabilités de Rennes, 1983

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