Estimation in Reversible Markov Chains

This article examines estimation of the one-step-ahead transition probabilities in a reversible Markov chain on a countable state space. A symmetrized moment estimator is proposed that exploits the reversible structure. Examples are given where the symmetrized estimator has superior asymptotic properties to those of a naive estimator, implying that knowledge of reversibility can sometimes improve estimation. The asymptotic mean and variance of the estimators are quantified. The results are proven using only elementary results such as the law of large numbers and the central limit theorem.