Ladder epochs and ladder chain of a Markov random walk with discrete driving chain

TL;DR

This paper investigates ladder epochs and chains in Markov random walks with discrete driving chains, establishing conditions for positive recurrence, stationary distributions, and illustrating differences between dual and original sequences.

Contribution

It extends existing results by providing conditions for ladder chain recurrence, explicit stationary distributions, and counterexamples using Palm duality and Wiener-Hopf factorization.

Findings

Ladder epochs are finite under dual sequence divergence.

The ladder chain is positive recurrent with an explicit stationary distribution.

Counterexample shows divergence of dual does not imply divergence of original sequence.

Abstract

Let $(M_{n},S_{n})_{n\ge 0}$ be a Markov random walk with positive recurrent driving chain $(M_{n})_{n\ge 0}$ having countable state space $\mathcal{S}$ and stationary distribution $\pi$. It is shown in this note that, if the dual sequence $({}^{\#}M_{n},{}^{\#}S_{n})_{n\ge 0}$ is positive divergent, i.e. ${}^{\#}S_{n}\to\infty$ a.s., then the strictly ascending ladder epochs $\sigma_{n}^{>}$ of $(M_{n},S_{n})_{n\ge 0}$ are a.s. finite and the ladder chain $(M_{\sigma_{n}^{>}})_{n\ge 0}$ is positive recurrent on some $\mathcal{S}^{>}\subset\mathcal{S}$. We also provide simple expressions for its stationary distribution $\pi^{>}$, an extension of the result to the case when $(M_{n})_{n\ge 0}$ is null recurrent, and a counterexample that demonstrates that ${}^{\#}S_{n}\to\infty$ a.s. does not necessarily entail $S_{n}\to\infty$ a.s., but rather $\limsup_{n\to\infty}S_{n}=\infty$ a.s. only. Our arguments are based on Palm duality theory, coupling and the Wiener-Hopf factorization for Markov random walks with discrete driving chain.