ON THE MARKOV RENEWAL THEOREM

[摘要] Let (S, J) be a measurable space with countably generated sigma-field J and (M(n), X(n))n greater-than-or-equal-to 0 a Markov chain with state space S x R and transition kernel P:S X (J x B) --> [0, 1]. Then (M(n), S(n))n greater-than-or-equal-to 0, where S(n) = X0 + ... + X(n) for n greater-than-or-equal-to 0, is called the associated Markov random walk. Markov renewal theory deals with the asymptotic behavior of suitable functionals of (M(n), S(n))n greater-than-or-equal-to 0 like the Markov renewal measure SIGMA(n greater-than-or-equal-to 0)P((M(n), S(n)) is-an-element-of A x (t + B)) as t --> infinity where A is-an-element-of J and B denotes a Borel subset of R. It is shown that the Markov renewal theorem as well as a related ergodic theorem for semi-Markov processes hold true if only Harris recurrence of (M(n))n greater-than-or-equal-to 0 is assumed. This was proved by purely analytical methods by Shurenkov [15] in the one-sided case where P(x, S X [0, infinity)) = 1 for all x is-an-element-of S. Our proof uses probabilistic arguments, notably the construction of regeneration epochs for (M(n))n greater-than-or-equal-to 0 such that (M(n), X(n))n greater-than-or-equal-to 0 is at least nearly regenerative and an extension of Blackwell's renewal theorem to certain random walks with stationary, 1-dependent increments.

[发布日期] 1994-03-01

[关键词] MARKOV RENEWAL THEORY;MARKOV RANDOM WALK;SEMI-MARKOV PROCESS;HARRIS RECURRENCE;REGENERATION EPOCHS;BLACKWELL RENEWAL THEOREM;RANDOM WALKS WITH STATIONARY, 1-DEPENDENT INCREMENTS;COUPLING