Hitting Time Distributions for Denumerable Birth and Death Processes

TL;DR

This paper extends Keilson's theorem to infinite state birth and death processes by deriving explicit Laplace transform formulas for hitting times, and applies these results to analyze convergence rates of ergodic processes.

Contribution

It provides the first explicit formulas for hitting times in infinite state birth and death processes, extending finite case results to countably infinite spaces.

Findings

Derived explicit Laplace transform formulas for hitting times.

Extended Keilson's theorem from finite to infinite state spaces.

Obtained explicit convergence rates for ergodic birth and death processes.

Abstract

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.