Stationary entrance chains and applications to random walks

Aleksandar Mijatović and Vladislav Vysotsky

Stochastic Processes and their Applications, 2025, vol. 188, issue C

Abstract: 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 Ac. Similarly, consider the exit chain, obtained by sampling Y at the exit times from Ac 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 Ac, 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.

Keywords: Level crossing; Random walk; Overshoot; Undershoot; Local time of random walk; Invariant measure; Stationary distribution; Entrance Markov chain; Exit Markov chain (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1016/j.spa.2025.104668

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