[Sur l'équation vectorielle stochastique
We study the behavior at infinity of the tail of the stationary solution of a multidimensional linear auto-regressive process with random coefficients. We exhibit an extended class of multiplicative coefficients satisfying a condition of irreducibility and proximality that yield to a heavy tail behavior.
On étudie le comportement à l'infini de la queue de la solution stationnaire d'un processus auto-régressif linéaire multidimensionnel à coefficients aléatoires. On donne une vaste classe de coefficients multiplicatifs vérifiant une condition d'irréductibilité et de proximalité qui conduisent à un comportement de type queue polynomiale.
Accepté le :
Publié le :
Benoîte de Saporta 1 ; Yves Guivarc'h 1 ; Emile Le Page 2
@article{CRMATH_2004__339_7_499_0,
author = {Beno{\^\i}te de Saporta and Yves Guivarc'h and Emile Le Page},
title = {On the multidimensional stochastic equation $ {Y}_{n+1}={A}_{n}{Y}_{n}+{B}_{n}$},
journal = {Comptes Rendus. Math\'ematique},
pages = {499--502},
publisher = {Elsevier},
volume = {339},
number = {7},
year = {2004},
doi = {10.1016/j.crma.2004.07.024},
language = {en},
}
TY - JOUR
AU - Benoîte de Saporta
AU - Yves Guivarc'h
AU - Emile Le Page
TI - On the multidimensional stochastic equation $ {Y}_{n+1}={A}_{n}{Y}_{n}+{B}_{n}$
JO - Comptes Rendus. Mathématique
PY - 2004
SP - 499
EP - 502
VL - 339
IS - 7
PB - Elsevier
DO - 10.1016/j.crma.2004.07.024
LA - en
ID - CRMATH_2004__339_7_499_0
ER -
Benoîte de Saporta; Yves Guivarc'h; Emile Le Page. On the multidimensional stochastic equation $ {Y}_{n+1}={A}_{n}{Y}_{n}+{B}_{n}$. Comptes Rendus. Mathématique, Volume 339 (2004) no. 7, pp. 499-502. doi : 10.1016/j.crma.2004.07.024. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.07.024/
[1] The stochastic equation
[2] Noncommuting random products, Trans. Amer. Math. Soc., Volume 108 (1963), pp. 377-428
[3] Boundary theory and stochastic processes on homogeneous spaces, Harmonic Analysis on Homogeneous Spaces, Proc. Sympos. Pure Math., vol. XXVI, American Mathematical Society, 1973, pp. 193-229
[4] Implicit renewal theory and tails of solutions of random equations, Ann. Appl. Probab., Volume 1 (1991), pp. 26-166
[5] Zariski closure and the dimension of the Gaussian law of the product of random matrices, Probab. Theory Related Fields, Volume 105 (1996), pp. 109-142
[6] Simplicité de spectres de Lyapunov et propriété d'isolation spectrale pour une famille d'opérateurs de transfert sur l'espace projectif (V. Kaimanovitch, ed.), Random Walks and Geometry, Workshop Vienna 2001, De Gruyter, 2004, pp. 181-259
[7] Products of random matrices: convergence theorems, Random matrices and their applications, Contemp. Math., Volume 50 (1986), pp. 31-54
[8] Random difference equations and renewal theory for products of random matrices, Acta Math., Volume 131 (1973), pp. 207-248
[9] Renewal theory for functionals of a Markov chain with general state space, Ann. Probab., Volume 2 (1974), pp. 355-386
[10] E. Le Page, Théorèmes de renouvellement pour les produits de matrices aléatoires. Equations aux différences aléatoires, Séminaires de probabilités de Rennes, 1983
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