Théorèmes limites pour les produits de matrices aléatoires Publications des séminaires de mathématiques et informatique de Rennes, 1980, fascicule 1 « Séminaire de probabilités », , exp. n o 4, p. 1-49 © Département de mathématiques et informatique, université de Rennes, 1980, tous droits réservés. L'accès aux archives de la série « Publications mathématiques et informatiques de Rennes » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

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International audienceWe consider a random walk on the affine group of the real line, we denote by P the corresponding Markov operator on R, and we study the Birkhoff sums associated with its trajectories. We show that, depending on the parameters of the random walk, the normalized Birkhoff sums converge in law to a stable law of exponent alpha is an element of]0, 2[ or to a normal law. The corresponding analysis is based on the spectral properties of two families of associated transfer operators P-t, T-t. The operator P-t is a Fourier operator and is considered here as a perturbation of the Markov operator P = P-0 of the random walk. The operator T-t is related to P-t by a symmetry of Heisenberg type and is also considered as a perturbation of the Markov operator T-0 = T. We prove that these operators have an isolated dominant eigenvalue which has an asymptotic expansion involving fractional powers of t. The parameters of this expansion have simple expressions in terms of tails and moments of the stationary measures of P and T

Consider a Markov chain (X n ) n 0 with values in the state space X. Let f be a real function on X and set S 0 = 0, S n = f (X 1 )+· · ·+f (X n ), n 1. Let P x be the probability measure generated by the Markov chain starting at X 0 = x. For a starting point y ∈ R denote by τ y the first moment when the Markov walk (y + S n ) n 1 becomes non-positive. Under the condition that S n has zero drift, we find the asymptotics of the probability P x (τ y > n) and of the conditional law P x ( y + S n · √ n | τ y > n ) as n → +∞.

Abstract. We prove an invariance principle for non-stationary random processes and establish a rate of convergence under a new type of mixing condition. The dependence is exponentially decaying in the gap between the past and the future and is controlled by an assumption on the characteristic function of the finite dimensional increments of the process. The distinct feature of the new mixing condition is that the dependence increases exponentially in the dimension of the increments. The proposed mixing property is particularly suited for processes whose behavior can be described in terms of spectral properties of some related family of operators. Several examples are discussed. We also work out explicit expressions for the constants involved in the bounds. When applied to Markov chains our result specifies the dependence of the constants on the properties of the underlying Banach space and on the initial state of the chain.