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Recurrence theorems for Markov random walks

Alsmeyer G.

Probability and Mathematical Statistics

2001

Vol. 21, Fasc. 1

123--134

Let (M,n, Sn)n≥0 be a Markov random walk whose driving chain (Mn)n≥0 with general state space (ℒ,Ϭ) is ergodic with unique stationary distribution ξ. Providing n−1 Sn→o in probability under Pξ, it is shown that the recurrence set of (Sn−γ(Mo) +γ(Mn))n≥o forms a closed subgroup of Rdepending on the lattice-type of (Mn, Sn)n≥o. The so-called shift function γ is bounded and appears in that lattice-type condition. The recurrence set of (Sn)n≥o itself is also given but may lookmore complicated depending on γ. The results extend the classical recurrence the orem for random walks with i.i.d. increments and further sharpenresults by Berbee, Dekking and others on the recurrence behavior of random walks with stationary increments.