On étudie la queue de la solution stationnaire de l'équation , , où 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 , , where the sequence 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.
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Benoîte de Saporta 1
@article{CRMATH_2005__340_1_55_0, author = {Beno{\^\i}te de Saporta}, title = {Tail of the stationary solution of the stochastic equation $ {Y}_{n+1}={a}_{n}{Y}_{n}+{b}_{n}$ with {Markovian} coefficients}, journal = {Comptes Rendus. Math\'ematique}, pages = {55--58}, publisher = {Elsevier}, volume = {340}, number = {1}, year = {2005}, doi = {10.1016/j.crma.2004.11.018}, language = {en}, }
TY - JOUR AU - Benoîte de Saporta TI - Tail of the stationary solution of the stochastic equation $ {Y}_{n+1}={a}_{n}{Y}_{n}+{b}_{n}$ with Markovian coefficients JO - Comptes Rendus. Mathématique PY - 2005 SP - 55 EP - 58 VL - 340 IS - 1 PB - Elsevier DO - 10.1016/j.crma.2004.11.018 LA - en ID - CRMATH_2005__340_1_55_0 ER -
%0 Journal Article %A Benoîte de Saporta %T Tail of the stationary solution of the stochastic equation $ {Y}_{n+1}={a}_{n}{Y}_{n}+{b}_{n}$ with Markovian coefficients %J Comptes Rendus. Mathématique %D 2005 %P 55-58 %V 340 %N 1 %I Elsevier %R 10.1016/j.crma.2004.11.018 %G en %F CRMATH_2005__340_1_55_0
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/
[1] Applied Probability and Queues, Wiley, Chichester, 1987
[2] The stochastic equation with stationary coefficients, Adv. Appl. Probab., Volume 18 (1986), pp. 211-220
[3] Implicit renewal theory and tails of solutions of random equations, Ann. Appl. Probab., Volume 1 (1991), pp. 26-166
[4] Products of random affine transformations, Lithuanian Math. J., Volume 20 (1980), pp. 279-282
[5] 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] Random difference equations and renewal theory for products of random matrices, Acta Math., Volume 131 (1973), pp. 207-248
[7] 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] Renewal theorem for a system of renewal equations, Ann. Inst. H. Poincare Probab. Statist., Volume 39 (2003), pp. 823-838
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