On longest consecutive patterns in Markov chains
Yizhou Xia · Stochastics · 2022
Consider a discrete-time homogeneous Markov chain with initial state i. We study the distribution of L(j,n), the length of the longest consecutive visits of this chain to state j until time n. We provide two limiting theorems for L(j,n) and establish asymptotics for the moment generating function of L(j,n). 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 L(j,n).