Conditioned limit theorems for hyperbolic dynamical systems
Abstract
Let $(\mathbb X, T)$ be a subshift of finite type equipped with the Gibbs measure $\nu$ and let $f$ be a realvalued Hölder continuous function on $\mathbb X$ such that $\nu(f) = 0$. Consider the Birkhoff sums $S_n f = \sum_{k=0}^{n1} f \circ T^{k}$, $n\geq 1$. For any $t \in \mathbb R$, denote by $\tau_t^f$ the first time when the sum $t+ S_n f$ leaves the positive halfline 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 \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 $\{ x \in \mathbb X: \tau_t^f(x)>n \}$.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.09838
 Bibcode:
 2021arXiv211009838G
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Probability
 EPrint:
 68 pages