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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {X_a } \right\}_{a \in Z_ + }$$\end{document} 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 Ln the length of the longest run of consecutive i’s for i ∈ T, that occurs in the sequence X1, ..., Xn. In this paper, we obtain a limit law and a week version of an Erdös-Rényi type law for Ln. A large deviation result of Ln is also discussed.