We proved the explicit formulas in Laplace transform of the hitting times for the birth and death processes on a denumerable state space with $\ift$ the exit or entrance boundary. This extends the well known Keilson's theorem from finite state space to infinite state space. We also apply these formulas to the fastest strong stationary time for strongly ergodic birth and death processes, and obtain the explicit convergence rate in separation.
Speed of coming down from infinity for birth and death processes
Quasi-stationary distribution for multi-dimensional birth and death processes conditioned to survival of all coordinates
On hitting times and fastest strong stationary times for skip-free and more general chains
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