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.
Publié le : 1990-10-14
Classification:
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,
60J10,
60G40
@article{1176990628,
author = {Diaconis, Persi and Fill, James Allen},
title = {Strong Stationary Times Via a New Form of Duality},
journal = {Ann. Probab.},
volume = {18},
number = {4},
year = {1990},
pages = { 1483-1522},
language = {en},
url = {http://dml.mathdoc.fr/item/1176990628}
}
Diaconis, Persi; Fill, James Allen. Strong Stationary Times Via a New Form of Duality. Ann. Probab., Tome 18 (1990) no. 4, pp. 1483-1522. http://gdmltest.u-ga.fr/item/1176990628/