Conditioned limit theorems for hyperbolic dynamical systems

TL;DR

This paper investigates the asymptotic behavior of conditioned sums in hyperbolic dynamical systems, establishing limit theorems analogous to those for random walks with i.i.d. increments.

Contribution

It introduces conditioned limit theorems for Birkhoff sums in hyperbolic systems, extending classical probabilistic results to deterministic dynamical contexts.

Findings

Asymptotic probabilities for the first exit time are derived.

Integral and local limit theorems are established under conditioning.

Results extend classical random walk theorems to 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 real-valued H\"older continuous function on $\mathbb X$ such that $\nu(f) = 0$. Consider the Birkhoff sums $S_n f = \sum_{k=0}^{n-1} f \circ T^{k}$, $n\geq 1$. For any $t \in \mathbb R$, denote by $\tau_t^f$ the first time when the sum $t+ S_n f$ leaves the positive 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 \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 $\{ x \in \mathbb X: \tau_t^f(x)>n \}$.