Conditioned limit theorems for products of random matrices

Abstract

Let g 1 , g 2 , . . . be i.i.d. random matrices in G L (d, R) . For any n ≥ 1 consider the product G n = g n . . . g 1 and the random processIt is well known that under appropriate assumptions, the sequence (log G n v ) n≥1 behaves like a sum of i.i.d. r.v.'s and satisfies standard classical properties such as the law of large numbers, the law of iterated logarithm and the central limit theorem. For any vector v with v > 1 denote by τ v the first time when the random process G n v enters the closed unit ball in R d . We establish the asymptotic as n → +∞ of the probability of the event {τ v > n} and find the limit law for the quantity 1 √ n log G n v conditioned that τ v > n.

The survival probability of a critical multi-type branching process in i.i.d. random environment

“…The asymptotic behaviour of the probability Px ,a (τ > n) is established in [13] when the matrices M n are invertible and in [21] when the M n has non negative entries, under several conditions P1-P5; the first step is to establish the existence of a P + -harmonic function h on X × R + . Our hypotheses H1, H2, H4 and H5 are exactly P1, P2, P4 and P5 in [21] and H3 is obviously stronger than P3.…”

“…where σ 2 > 0 is the variance of the Markov walk (S n ) n≥0 , given in [13]. Moreover, there exists a constant c > 0 such that for anyx ∈ X, a ≥ 0 and…”

“…The Markov walk (X n , S n ) n≥0 has been studied by many authors (see for instance [9], [14] or [13]). All the works are based on the fact that the transition kernel of the chain (X n ) n≥0 has some "nice" spectral properties, namely its restriction to the space of Lipschitz functions on X is quasicompact.…”

“…To tackle the "general" case when the action of the random matrices M k is strongly irreducible (see hypothesis H2 below), limit theorems on the fluctuations of the norm of products of random matrices were required; they have been recently achieved in [12] and [13] with asymptotic results on the tails of certain hitting time distribution.…”

First-passage times for random walks with nonidentically distributed increments

“…Under assumptions of Theorem 1 asymptotics (11) hold with function U g defined in(13) which is slowly varying. Moreover, for all n ≥ 1α * n := B n P(T g > n) EZ * n − 2ϕ(0) ≤ C 1 ρ 2/for some C 1 < ∞.We split the proof into several steps.…”