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

摘要

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