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Stationary entrance chains and applications to random walks

Aleksandar Mijatovic, Vladislav Vysotsky

TL;DR

The paper develops a unified, induction-based framework for analyzing entrance and exit Markov chains sampled from a base chain Y on a Polish space. By leveraging inducing and duality, it yields explicit invariant measures for the entrance and exit chains and clarifies conditions for their existence, ergodicity, and uniqueness, even beyond recurrence (including certain transient cases). Applied to random walks in $\mathbb{R}^d$, the authors obtain explicit stationary measures for entrance chains into subdomains such as orthants, with dimension-one results tying overshoots to local time and level-crossings to a CLT. The work connects infinite ergodic theory, duality, and Kac-type formulas to provide concrete tools for stability and limit theorems in Markov chains and random walks, including applications to level-crossing processes and related switching/renewal dynamics.

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 $A^c$. 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.

Stationary entrance chains and applications to random walks

TL;DR

The paper develops a unified, induction-based framework for analyzing entrance and exit Markov chains sampled from a base chain Y on a Polish space. By leveraging inducing and duality, it yields explicit invariant measures for the entrance and exit chains and clarifies conditions for their existence, ergodicity, and uniqueness, even beyond recurrence (including certain transient cases). Applied to random walks in , the authors obtain explicit stationary measures for entrance chains into subdomains such as orthants, with dimension-one results tying overshoots to local time and level-crossings to a CLT. The work connects infinite ergodic theory, duality, and Kac-type formulas to provide concrete tools for stability and limit theorems in Markov chains and random walks, including applications to level-crossing processes and related switching/renewal dynamics.

Abstract

For a Markov chain with values in a Polish space, consider the entrance chain, obtained by sampling at the moments when it enters a fixed set from its complement . Similarly, consider the exit chain, obtained by sampling at the exit times from to . 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 has a known invariant measure. We give explicit formulas for invariant measures of the entrance and exit chains under certain recurrence-type assumptions on and , which apply even for transient chains. Then we study uniqueness and ergodicity of these invariant measures assuming that is topologically recurrent, topologically irreducible, and weak Feller. We give applications to random walks in , 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 and . 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.
Paper Structure (13 sections, 23 theorems, 119 equations)

This paper contains 13 sections, 23 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Entrance and exit Markov chains
  4. Invariance by inducing for recurrent chains
  5. Setup and notation
  6. General recurrent Markov chains
  7. Weak Feller recurrent Markov chains
  8. Invariance by duality
  9. Applications to random walks in $\mathbb{R}^d$
  10. The number of level-crossings for one-dimensional random walks
  11. Limit theorem
  12. Stationarity of level-crossings
  13. Induced transformations in infinite ergodic theory

Key Result

Lemma 2.1

Let $Y$ be a Markov chain that take values in a measurable space $(\mathcal{X}, \mathcal{F})$, and let $A \in \mathcal{F}$. Then for every $x_0 \in \mathcal{X}$, the entrance sequence $Y^{\rangle A}$ and the exit sequence $Y^{A^c \rangle}$ are time-homogeneous Markov chains under $\mathbb{P}_{x_0}$ and If $x_0 \in N_A$, these chains take values in the sets $A \cap N_A$ and $A^c_{ex} \cap N_A$, r

Theorems & Definitions (44)

  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • Corollary 3.1
  • proof
  • Remark 3.1
  • proof : Proof of Theorem \ref{['thm: inducing MC recurrent']}
  • Theorem 3.2
  • ...and 34 more