Strong Stationary Times Via a New Form of Duality

Abstract

A strong stationary time for a Markov chain (X,) is a stopping time T for which XT is stationary and independent of T. Such times yield sharp bounds on certain measures of nonstationarity for X at fixed finite times n . We construct an absorbing dual Markov chain with absorption time a strong stationary time for X. We relate our dual to a notion of duality used in the study of interacting particle systems. For birth and death chains, our dual is again birth and death and permits a stochastic interpretation of the eigenvalues of the transition matrix for X. The duality approach unifies and extends the analysis of previous constructions and provides several new examples. 1. Overview. 1.1. Introduction. Strong stationary times give a probabilistic approach to bounding speed of convergence to stationarity for Markov chains. They were introduced (under the name strong uniform times) by Aldous and Diaconis (1986), who give a number of examples demonstrating both sharp bounds and successful analysis of problems not amenable to other techniques such as eigenvalues and coupling. Diaconis [(1988), Chapter 41 and Matthews (1987, 1988) construct strong stationary times for various random walks on groups. Aldous and Diaconis (1987) develop some basic theory, showing that an optimal strong stationary time exists for any ergodic chain, i.e., one that is irreducible, positive recurrent and aperiodic. Closely related constructions appear in Brown (1975), Athreya and Ney (1978) and Nummelin (1986). Thorisson (1988) discusses connections with coupling. In this paper we extend the notion of strong stationary time to that of strong stationary duality for discrete time, finite state Markov chains. Diaco-nis and Fill (1990) treat problems with countably infinite state space. Fill (1990a, b) treats continuous time chains and Fill (1990~) applies strong sta-tionary duality to diffusions. We begin with a simple example. EXAMPLE 1.1. Simple symmetric random walk on a d-point circle. Let Zd be the integers modulo d , regarded as d labelled points arranged about a circle. A random walk starts a t 0 and with probability 1/3 each moves one step in either direction along the circle or remains fixed. The stationary distribution