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

de Saporta, Benoîte

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
[Queue de la solution stationnaire de l'équation

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

à coefficients markoviens]

Présenté par : Marc Yor

de Saporta, Benoîte ¹

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

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

Résumé
Abstract

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.

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.

Reçu le : 2004-01-08
Accepté le : 2004-11-03
Publié le : 2004-12-19

DOI : 10.1016/j.crma.2004.11.018

Affiliations des auteurs :

de Saporta, Benoîte ¹

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

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     title = {Tail of the stationary solution of the stochastic equation $ {Y}_{n+1}={a}_{n}{Y}_{n}+{b}_{n}$ with {Markovian} coefficients},
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de Saporta, Benoîte. Tail of the stationary solution of the stochastic equation $ {Y}_{n+1}={a}_{n}{Y}_{n}+{b}_{n}$ with Markovian coefficients. Comptes Rendus. Mathématique, Tome 340 (2005) no. 1, pp. 55-58. doi : 10.1016/j.crma.2004.11.018. http://www.numdam.org/articles/10.1016/j.crma.2004.11.018/

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