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Strong stationary duality for continuous-time Markov chains. I: Theory. (English) Zbl 0746.60075

Let \(X\equiv (X(t),\;0\leq t<\infty)\) be an ergodic continuous-time Markov chain with finite or countably infinite state space \(S\), distribution \(\pi_ t\) at time \(t\) and stationary distribution \(\pi\). The variation distance is defined as \(\| \pi_ t - \pi\|=\sup_{A\subset S}|\pi_ t(A) - \pi(A)|\), and a strong stationary time \(T\) is a randomized stopping time for \(X\) such that, conditionally on \((T<\infty)\), \(X(T)\) has distribution \(\pi\) and is independent of \(T\). The author shows that strong stationary times lead to bounds on variation distance, and that they can be built by constructing and analyzing a strong stationary dual Markov chain. A particularly simple construction is given for the special class of monotone likelihood chains, which incorporates birth-death processes.

MSC:

60J27 Continuous-time Markov processes on discrete state spaces
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