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

2015/11/17 by Alsmeyer, Gerold
#60J10 #60K15 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1511.05361

Abstract

Let (M_n,S_n)_{n≥ 0} be a Markov random walk with positive recurrent driving chain (M_n)_{n≥ 0} having countable state space S and stationary distribution π. It is shown in this note that, if the dual sequence (^#M_n,^#S_n)_{n≥ 0} is positive divergent, i.e. ^#S_n→∞ a.s., then the strictly ascending ladder epochs σ_n^> of (M_n,S_n)_{n≥ 0} are a.s. finite and the ladder chain (M_σ_n^>)_{n≥ 0} is positive recurrent on some S^>\subsetS. We also provide simple expressions for its stationary distribution π^>, an extension of the result to the case when (M_n)_{n≥ 0} is null recurrent, and a counterexample that demonstrates that ^#S_n→∞ a.s. does not necessarily entail S_n→∞ a.s., but rather \limsup_n→∞S_n=∞ 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.

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