CONDITIONED LIMIT THEOREMS FOR HYPERBOLIC DYNAMICAL SYSTEMS
Résumé
Let $(\bb X, T)$ be a subshift of finite type equipped with the Gibbs measure $\nu$ and let
$f$ be a real-valued H\"older continuous function on $\bb X$ such that $\nu(f) = 0$.
Consider the Birkhoff sums $S_n f = \sum_{k=0}^{n-1} f \circ T^{k}$, $n\geq 1$.
For any $t \in \bb R$, denote by $\tau_t^f$ the first time when the sum $t+ S_n f$
leaves the positive half-line for some $n\geq 1$.
By analogy with the case of random walks with independent identically distributed increments,
we study the asymptotic as $ n\to\infty $ of the probabilities
$ \nu(x\in \bb X: \tau_t^f(x)>n) $ and $ \nu(x\in \bb X: \tau_t^f(x)=n) $.
We also establish integral and local type limit theorems for the sum $t+ S_n f(x)$
conditioned on the set $\{ x \in \bb X: \tau_t^f(x)>n \}.$
Origine : Fichiers produits par l'(les) auteur(s)