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

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Fill, James Allen

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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.

Comment: To appear in Journal of Theoretical Probability. Main change with this version: small corrections to final section

Comment: To appear in Journal of Theoretical Probability. Main change with this version: small corrections to final section

##### Keywords

Mathematics - Probability, 60J25 (Primary), 60J35, 60J10, 60G40 (Secondary)