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
[Queue de la solution stationnaire de l'équation

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

à coefficients markoviens]

Présenté par : 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.

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 :

Benoîte de Saporta 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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     pages = {55--58},
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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/

Bibliographie
Cité par

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

[4] A.K. Grincevičius Products of random affine transformations, Lithuanian Math. J., Volume 20 (1980), pp. 279-282

[5] J.D. Hamilton Estimation, inference and forecasting of time series subject to change in regime (G. Maddala; C.R. Rao; D.H. Vinod, eds.), Handbook of Statistics, vol. 11, 1993, pp. 230-260

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

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

[8] C. Klüppelberg, S. Pergamenchtchikov, The tail of the stationary distribution of a random coefficient AR(q) model, preprint, 2002

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

[10] B. de Saporta Renewal theorem for a system of renewal equations, Ann. Inst. H. Poincare Probab. Statist., Volume 39 (2003), pp. 823-838

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