Probability Theory
Tail of the stationary solution of the stochastic equation Yn+1=anYn+bn with Markovian coefficients
Comptes Rendus. Mathématique, Volume 340 (2005) no. 1, pp. 55-58.

In this Note, we deal with the real stochastic difference equation Yn+1=anYn+bn, nZ, where the sequence (an) 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 Yn+1=anYn+bn, nZ, où (an) 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.

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DOI: 10.1016/j.crma.2004.11.018
de Saporta, Benoîte 1

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

[1] Asmussen, S. Applied Probability and Queues, Wiley, Chichester, 1987

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

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

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

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

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

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

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