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Contents of PMS, Vol. 21, Fasc. 1,
pages 123 - 134
 

RECURRENCE THEOREMS FOR MARKOV RANDOM WALKS

Gerold Alsmeyer

Abstract: Let (M ,S ) n nn>0 be a Markov random walk whose driving chain (M ) n n>0 with general state space (S,C) is ergodic with unique stationary distribution q. Providing n- 1S -- > 0 n in probability under P , q it is shown that the recurrence set of (S - g(M )+ g(M )) n 0 n n>0 forms a closed subgroup of R depending on the lattice-type of (M ,S ) . n n n>0 The so-called shift function g is bounded and appears in that lattice-type condition. The recurrence set of (S ) n n>0 itself is also given but may look more complicated depending on g. The results extend the classical recurrence theorem for random walks with i.i.d. increments and further sharpen results by Berbee, Dekking and others on the recurrence behavior of random walks with stationary increments.

1991 AMS Mathematics Subject Classification: 60J05, 60J15, 60K05, 60K15.

Key words and phrases: Markov random walk, random walk with stationary increments, recurrence point.

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