Conditioned limit theorems for hyperbolic dynamical systems

Abstract

AbstractLet $({\mathbb X}, T)$ be a subshift of finite type equipped with the Gibbs measure $\nu $ and let f be a real-valued Hölder continuous function on ${\mathbb X}$ such that $\nu (f) = 0$ . Consider the Birkhoff sums $S_n f = \sum _{k=0}^{n-1} f \circ T^{k}$ , $n\geqslant 1$ . For any $t \in {\mathbb R}$ , denote by $\tau _t^f$ the first time when the sum $t+ S_n f$ leaves the positive half-line for some $n\geqslant 1$ . By analogy with the case of random walks with independent and identically distributed increments, we study the asymptotic as $ n\to \infty $ of the probabilities $ \nu (x\in {\mathbb X}: \tau _t^f(x)>n) $ and $ {\nu (x\in {\mathbb 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 $\{ x \in {\mathbb X}: \tau _t^f(x)>n \}.$