Tail of the stationary solution of the stochastic equation $ {Y}_{n+1}={a}_{n}{Y}_{n}+{b}_{n}$ with Markovian coefficients

Benoîte de Saporta

doi:10.1016/j.crma.2004.11.018

Probability Theory

Tail of the stationary solution of the stochastic equation

Y_{n + 1} = a_{n} Y_{n} + b_{n}

with Markovian coefficients

Presented by: Marc Yor

Benoîte de Saporta ¹

¹ IRMAR, université de Rennes I, campus de Beaulieu, 35042 Rennes cedex, France

Comptes Rendus. Mathématique, Volume 340 (2005) no. 1, pp. 55-58.

Abstract
Résumé

In this Note, we deal with the real stochastic difference equation $Y_{n + 1} = a_{n} Y_{n} + b_{n}$ , $n \in Z$ , where the sequence $(a_{n})$ is a finite state space Markov chain. By means of the renewal theory, we give a precise description of the situation where the tail of its stationary solution exhibits power law behavior.

On étudie la queue de la solution stationnaire de l'équation $Y_{n + 1} = a_{n} Y_{n} + b_{n}$ , $n \in Z$ , où $(a_{n})$ est une chaîne de Markov à espace d'états fini. Par des méthodes de renouvellement, on donne une caractérisation détaillée du cas où la queue est polynômiale.

Received: 2004-01-08
Accepted: 2004-11-03
Published online: 2004-12-19

DOI: 10.1016/j.crma.2004.11.018

Author's affiliations:

Benoîte de Saporta ¹

¹ IRMAR, université de Rennes I, campus de Beaulieu, 35042 Rennes cedex, France

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Benoîte de Saporta. Tail of the stationary solution of the stochastic equation $ {Y}_{n+1}={a}_{n}{Y}_{n}+{b}_{n}$ with Markovian coefficients. Comptes Rendus. Mathématique, Volume 340 (2005) no. 1, pp. 55-58. doi : 10.1016/j.crma.2004.11.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.11.018/

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Cited by

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