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Conditioned limit theorems for hyperbolic dynamical systems

Part of: Markov processes Limit theorems Ergodic theory Smooth dynamical systems: general theory

Published online by Cambridge University Press:  20 March 2023

ION GRAMA Open the ORCID record for ION GRAMA [Opens in a new window] ,
JEAN-FRANÇOIS QUINT  and
HUI XIAO Open the ORCID record for HUI XIAO [Opens in a new window]
ION GRAMA*
Affiliation:
Université de Bretagne Sud, CNRS UMR 6205 LMBA, Campus de Tohannic 56017, Vannes, France
JEAN-FRANÇOIS QUINT
Affiliation:
Université de Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33405 Talence, France (e-mail: jean-francois.quint@math.u-bordeaux.fr)
HUI XIAO
Affiliation:
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China (e-mail: xiaohui@amss.ac.cn)
*
e-mail: ion.grama@univ-ubs.fr
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Abstract

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 $\{ x \in {\mathbb X}: \tau _t^f(x)>n \}.$

Keywords

conditioned limit theoremsconditioned local limit theoremsexit timedynamical systems

MSC classification

Primary: 37A30: Ergodic theorems, spectral theory, Markov operators 37A50: Relations with probability theory and stochastic processes 37C05: Smooth mappings and diffeomorphisms
Secondary: 60F05: Central limit and other weak theorems 60J05: Discrete-time Markov processes on general state spaces
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 44 , Issue 1 , January 2024 , pp. 50 - 117
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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