Stationary entrance chains and applications to random walks
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arXiv
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| Format: | Preprint |
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2024
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| author | Mijatovic, Aleksandar Vysotsky, Vladislav |
| author_facet | Mijatovic, Aleksandar Vysotsky, Vladislav |
| contents | For a Markov chain $Y$ with values in a Polish space, consider the entrance chain, obtained by sampling $Y$ at the moments when it enters a fixed set $A$ from its complement $A^c$. Similarly, consider the exit chain, obtained by sampling $Y$ at the exit times from $A^c$ to $A$. We use the method of inducing from ergodic theory to study invariant measures of these two types of Markov chains in the case when the initial chain $Y$ has a known invariant measure. We give explicit formulas for invariant measures of the entrance and exit chains under certain recurrence-type assumptions on $A$ and $A^c$, which apply even for transient chains. Then we study uniqueness and ergodicity of these invariant measures assuming that $Y$ is topologically recurrent, topologically irreducible, and weak Feller.
We give applications to random walks in $R^d$, which we regard as ``stationary'' Markov chains started under the Lebesgue measure. We are mostly interested in dimension one, where we study the Markov chain of overshoots above the zero level of a random walk that oscillates between $-\infty$ and $+\infty$. We show that this chain is ergodic, and use this result to prove a central limit theorem for the number of level crossings of a random walk with zero mean and finite variance of increments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00619 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stationary entrance chains and applications to random walks Mijatovic, Aleksandar Vysotsky, Vladislav Probability Primary: 60J10, 60G50, 37A50, secondary: 60J55, 60G10, 60G40, 60F05, 28D05 For a Markov chain $Y$ with values in a Polish space, consider the entrance chain, obtained by sampling $Y$ at the moments when it enters a fixed set $A$ from its complement $A^c$. Similarly, consider the exit chain, obtained by sampling $Y$ at the exit times from $A^c$ to $A$. We use the method of inducing from ergodic theory to study invariant measures of these two types of Markov chains in the case when the initial chain $Y$ has a known invariant measure. We give explicit formulas for invariant measures of the entrance and exit chains under certain recurrence-type assumptions on $A$ and $A^c$, which apply even for transient chains. Then we study uniqueness and ergodicity of these invariant measures assuming that $Y$ is topologically recurrent, topologically irreducible, and weak Feller. We give applications to random walks in $R^d$, which we regard as ``stationary'' Markov chains started under the Lebesgue measure. We are mostly interested in dimension one, where we study the Markov chain of overshoots above the zero level of a random walk that oscillates between $-\infty$ and $+\infty$. We show that this chain is ergodic, and use this result to prove a central limit theorem for the number of level crossings of a random walk with zero mean and finite variance of increments. |
| title | Stationary entrance chains and applications to random walks |
| topic | Probability Primary: 60J10, 60G50, 37A50, secondary: 60J55, 60G10, 60G40, 60F05, 28D05 |
| url | https://arxiv.org/abs/2403.00619 |