## 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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### References:

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