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Conditioned limit theorems for hyperbolic dynamical systems
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 44 / Issue 1 / January 2024
- Published online by Cambridge University Press:
- 20 March 2023, pp. 50-117
- Print publication:
- January 2024
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- Article
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Let
$({\mathbb X}, T)$ 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 
$\{ x \in {\mathbb X}: \tau _t^f(x)>n \}.$
