The Annals of Probability

Strong Stationary Times Via a New Form of Duality

Persi Diaconis and James Allen Fill

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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

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

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Markov chains rates of convergence stochastic monotonicity monotone likelihood ratio birth and death chains eigenvalues random walk Ehrenfest chain strong stationary duality dual processes Siegmund duality time reversal Doob $H$ transform total variation


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.

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