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October, 1990 Strong Stationary Times Via a New Form of Duality
Persi Diaconis, James Allen Fill
Ann. Probab. 18(4): 1483-1522 (October, 1990). DOI: 10.1214/aop/1176990628


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.


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Persi Diaconis. James Allen Fill. "Strong Stationary Times Via a New Form of Duality." Ann. Probab. 18 (4) 1483 - 1522, October, 1990.


Published: October, 1990
First available in Project Euclid: 19 April 2007

zbMATH: 0723.60083
MathSciNet: MR1071805
Digital Object Identifier: 10.1214/aop/1176990628

Primary: 60J10
Secondary: 60G40

Keywords: birth and death chains , Doob $H$ transform , Dual processes , Ehrenfest chain , Eigenvalues , Markov chains , monotone likelihood ratio , Random walk , rates of convergence , Siegmund duality , Stochastic monotonicity , strong stationary duality , Time reversal , Total variation

Rights: Copyright © 1990 Institute of Mathematical Statistics


Vol.18 • No. 4 • October, 1990
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