## The Annals of Probability

### 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.

#### Article information

Source
Ann. Probab. Volume 18, Number 4 (1990), 1483-1522.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176990628

Digital Object Identifier
doi:10.1214/aop/1176990628

Mathematical Reviews number (MathSciNet)
MR1071805

Zentralblatt MATH identifier
0723.60083

JSTOR
links.jstor.org

#### Citation

Diaconis, Persi; Fill, James Allen. Strong Stationary Times Via a New Form of Duality. Ann. Probab. 18 (1990), no. 4, 1483--1522. doi:10.1214/aop/1176990628. https://projecteuclid.org/euclid.aop/1176990628