Some results associated with the longest run in a strongly ergodic Markov chain

This paper discusses the asymptotic behaviors of the longest run on a countable state Markov chain. Let {Xa}a.Z+ be a stationary strongly ergodic reversible Markov chain on countablestate space S = {1, 2, ...}. Let T . S be an arbitrary finite subset of S. Denote by L n the length of the longest run of consecutive i.s for i . T, that occurs in the sequence X 1, ..., X n . In this paper, we obtain a limit law and a week version of an Erdös-Rényi type law for L n . A large deviation result of L n is also discussed.