On longest consecutive patterns in Markov chains

Abstract

ABSTRACT Consider a discrete-time homogeneous Markov chain with initial state i. We study the distribution of , the length of the longest consecutive visits of this chain to state j until time n. We provide two limiting theorems for and establish asymptotics for the moment generating function of . We conclude by closing the open problem raised by the authors of [T. Konstantopoulos, Z. Liu, and X. Yang, Laplace transform asymptotics and large deviation principles for longest success runs in Bernoulli trials, J. Appl. Probab. 53 (2016), pp. 747–764.] by providing two large deviation principles for .