Conditioned limit theorems for hyperbolic dynamical systems

Abstract

Let (X, T ) be a subshift of finite type equipped with the Gibbs measure ν and let f be a real-valued Hölder continuous function on X such that ν(f1. For any t ∈ R, denote by τ f t the first time when the sum t + S n f leaves the positive half-line for some n 1. By analogy with the case of random walks with independent identically distributed increments, we study the asymptotic as n → ∞ of the probabilities ν(x ∈ X :We also establish integral and local type limit theorems for the sum t+S n f (x) conditioned on the set {x ∈ X : τ f t (x) > n}.

Berry-Esseen bounds and moderate deviations for the norm, entries and spectral radius of products of positive random matrices

“…where ν is the unique invariant measure defined by (2.1). The following lemma is from [20,Lemma 5.3].…”

“…By using the spectral gap theory (cf. [26,27]) for products of positive random matrices, the proof of Theorem 3.2 can be carried out in an analogous way as those in [22,20] and hence the details are omitted.…”

“…For additive functionals of finite state Markov chains, the exact asymptotic has been established in [18]. Recent progress in this direction for hyperbolic dynamical systems has been made by Grama, Quint and Xiao [20]. Let us stress that for products of random matrices it is still an open question how to obtain the exact asymptotic of order n −3/2 .…”

“…Unfortunately, the method of the present paper does not allow to reach this. We refer to the paper [20], where we develop a new approach for proving the exact asymptotics of type (1.5) in the case of hyperbolic dynamical systems.…”

“…We follow the approach developed in [22], but compared with the case of independent random variables, the situation here is much more delicate. We have to adapt the techniques from [20] developed in the context of hyperbolic dynamical systems. The idea is to make use of the Markov property to write the local probability (1.2) as a limit convolution of two main terms, one of them corresponding to an ordinary local limit theorem, while the other corresponds to a conditioned integral limit theorem.…”