Etude de la solution stationnaire de l'équation Y(n+1)=a(n)Y(n)+b(n) à coefficients aléatoires

Le modele auto-regressif lineaire (AR) en temps discret et a coefficients aleatoires englobe de nombreuses classes de modeles tres utilises en modelisation statistique. Sous des hypotheses simples, ce modele a une unique solution stationnaire. Le comportement a l'infini de sa queue a ete etudie par H. Kesten, E. LePage puis C. Goldie lorsque les coefficients sont independants. Cette these etend leurs resultats dans deux directions. Dans une premiere partie, on etudie le modele AR scalaire a regime markovien introduit par J. D. Hamilton en econometrie. On obtient un resultat similaire au cas independant qui s'etend aussi au temps continu. Dans une deuxieme partie, on s'interesse au modele multidimensionnel a coefficient independants. On etend les resultats existants a une vaste classe de coefficients verifiant une condition d'irreductibilite et de proximalite. Les techniques utilisees dans les deux parties font appel a la theorie du renouvellement et des operateurs markoviens.

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