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Strong stationary times via a new form of duality. (English) Zbl 0723.60083
The very interesting notion of SST (strong stationary time) of D. Aldous and P. Diaconis [Adv. Appl. Math. 8, 69-97 (1987; Zbl 0631.60065)] is extended to that of strong stationary duality for finite Markov chains (MC) ${X}_{n}$. Then the problem of analyzing convergence to stationarity is turned into a study of first passage times; it yields sharp bounds of the variation distance between the measure of ${X}_{n}$ and the stationary measure $\pi$. A general and often practical method for ergodic MC to construct SST is obtained. The chains with monotone likelihood ratio (especially death-and-birth chains), of which duals are particularly simply constructed, are discussed in detail with concrete examples. The relation of duals here to the notions of duality used by Liggett and Siegmund are considered.
Reviewer: M.Qian (Beijing)

##### MSC:
 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60G40 Stopping times; optimal stopping problems; gambling theory