A central limit theorem for absorbing Markov chains

Abstract

SUMMARY A central limit theorem is obtained for a sequence of random variables defined on a finite absorbing Markov chain, conditional on absorption not having taken place. The transition count for such a chain when suitably scaled is found to follow a multivariate normal distribution asymptotically. In the case where the transition probability matrix of the chain is a function of a single parameter ac, a consistent estimator for oc is found; this estimator is asymptotically normally distributed about the true value of c. The result is illustrated by a simulation study of a genetic model of Moran, for the case of one-way mutation in a population of gametes. Absorbing Markov chains appear frequently in the literature as models for processes in medicine and biology, where, for example, an absorbing state may represent the disappearance of a certain characteristic in the population. Bartlett (1960) pointed out that in many cases the time to reach an absorbing state is effectively infinite, and in such cases it is relevant to consider the behaviour of the process within the transient states. This leads to the study of chains in which it is known that absorption has not taken place at a certain stage, and to the theory of quasi-stationary distributions as developed by Darroch & Seneta (1965). In this paper we consider the behaviour of a finite absorbing Markov chain, conditional on the knowledge that an absorbing state has not been reached. In ? 2 a central limit theorem for processes defined on such a chain is derived. This theorem provides the basis for the theory of statistical inference which is developed in ? 3. The results are illustrated in ? 4 by a genetic model of Moran.