Mathematical Sciences

This dissertation describes the research that we have done concerning reversible Markov chains. We first present definitions for what it means for a Markov chain to be reversible. We then give applications of where reversible Markov chains are used and give a brief history of Markov chain inference. Finally, two journal articles are found in the paper, one that is already published and another which is currently being submitted. The first article examines estimation of the one-step-ahead transition probabilities in a re-versible 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 reversibil-ity 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. The second article introduces two statistics that assess whether (or not) a sequence sampled from a time-homogeneous Markov chain on a finite state space is reversible. The test statistics ar