Fluctuation Theory for Markov Random Walks

Alsmeyer G., Buckmann F.

Abstract

Two fundamental theorems by Spitzer–Erickson and Kesten–Maller on the fluctuation-type (positive divergence, negative divergence or oscillation) of a real-valued random walk (Formula presented.) with iid increments (Formula presented.) and the existence of moments of various related quantities like the first passage into (Formula presented.) and the last exit time from (Formula presented.) for arbitrary (Formula presented.) are studied in the Markov-modulated situation when the (Formula presented.) are governed by a positive recurrent Markov chain (Formula presented.) on a countable state space (Formula presented.); thus, for a Markov random walk (Formula presented.). Our approach is based on the natural strategy to draw on the results in the iid case for the embedded ordinary random walks (Formula presented.), where (Formula presented.) denote the successive return times of M to state i, and an analysis of the excursions of the walk between these epochs. However, due to these excursions, generalizations of the aforementioned theorems are surprisingly more complicated and require the introduction of various excursion measures so as to characterize the existence of moments of different quantities.