The passage time distribution for a birth-and-death chain: Strong stationary duality gives a first stochastic proof

TL;DR

This paper provides a novel probabilistic proof for the distribution of passage times in birth-and-death chains using strong stationary duality, linking eigenvalues to distribution parameters and offering explicit constructions.

Contribution

It introduces the first stochastic proof for passage time distributions in discrete and continuous birth-and-death chains, connecting eigenvalues to distribution parameters.

Findings

Probabilistic proof of passage time distribution using strong stationary duality.

Explicit construction of the sum of independent exponential variables for passage times.

Connection of eigenvalues to distribution parameters and proof of Ray-Knight theorem.

Abstract

A well-known theorem usually attributed to Keilson states that, for an irreducible continuous-time birth-and-death chain on the nonnegative integers and any d, the passage time from state 0 to state d is distributed as a sum of d independent exponential random variables. Until now, no probabilistic proof of the theorem has been known. In this paper we use the theory of strong stationary duality to give a stochastic proof of a similar result for discrete-time birth-and-death chains and geometric random variables, and the continuous-time result (which can also be given a direct stochastic proof) then follows immediately. In both cases we link the parameters of the distributions to eigenvalue information about the chain. We also discuss how the continuous-time result leads to a proof of the Ray-Knight theorem. Intimately related to the passage-time theorem is a theorem of Fill that any fastest strong stationary time T for an ergodic birth-and-death chain on {0, >..., d} in continuous time with generator G, started in state 0, is distributed as a sum of d independent exponential random variables whose rate parameters are the nonzero eigenvalues of the negative of G. Our approach yields the first (sample-path) construction of such a T for which individual such exponentials summing to T can be explicitly identified.