Reversibility of first-order autoregressive processes

Abstract

The present paper deals with reversibility of autoregressive processes of first order, namely AR(1). Reversibility of Markov chains with general state space was studied by Ōsawa (1985). His results are applied to AR(1) processes in this paper. Consider a Markov chain with transition densities whose state space is a Euclidian space. For reversibility of a Markov chain two ideas are available. One is reversibility with respect to some density, and the other time-reversibility. A time-reversibility chain has a stationary distribution constructed by the reversible density. On the other hand a reversible chain does not necessarily have one. We first state general theorems which provide criteria for determining whether a Markov chain is reversible with respect to some density, time-reversible or not. These are applied to AR(1) processes on the real line. We shall find some examples of Markov chains which have a reversible density but cannot be time-reversible. Further, we obtain a necessary and sufficient condition under which a certain multivariate AR(1) process is reversible.