Conditioned limit theorems for hyperbolic dynamical systems
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866913620068139008 |
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| author | Grama, Ion Quint, Jean-François Xiao, Hui |
| author_facet | Grama, Ion Quint, Jean-François Xiao, Hui |
| contents | Let $(\mathbb X, T)$ be a subshift of finite type equipped with the Gibbs measure $ν$ and let $f$ be a real-valued Hölder continuous function on $\mathbb X$ such that $ν(f) = 0$. 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 $τ_t^f$ 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 $ ν(x\in \mathbb X: τ_t^f(x)>n) $ and $ ν(x\in \mathbb X: τ_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 $\{ x \in \mathbb X: τ_t^f(x)>n \}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_09838 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Conditioned limit theorems for hyperbolic dynamical systems Grama, Ion Quint, Jean-François Xiao, Hui Dynamical Systems Probability Let $(\mathbb X, T)$ be a subshift of finite type equipped with the Gibbs measure $ν$ and let $f$ be a real-valued Hölder continuous function on $\mathbb X$ such that $ν(f) = 0$. 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 $τ_t^f$ 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 $ ν(x\in \mathbb X: τ_t^f(x)>n) $ and $ ν(x\in \mathbb X: τ_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 $\{ x \in \mathbb X: τ_t^f(x)>n \}$. |
| title | Conditioned limit theorems for hyperbolic dynamical systems |
| topic | Dynamical Systems Probability |
| url | https://arxiv.org/abs/2110.09838 |