Strong Stationary Times Via a New Form of Duality

Abstract

A strong stationary time for a Markov chain $(X_n)$ is a stopping time $T$ for which $X_T$ 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.