On hitting times and fastest strong stationary times for skip-free and more general chains

TL;DR

This paper provides new, simpler proofs and explicit representations for hitting times and fastest strong stationary times in skip-free Markov chains, extending results to more general chains and discrete-time cases.

Contribution

The paper introduces simpler proofs and explicit sum-of-exponentials representations for hitting times and stationary times in skip-free chains, including discrete-time and more general chains.

Findings

Explicit sum of exponentials for hitting times in skip-free chains

Simpler proof techniques compared to previous work

Extensions to discrete-time and more general chains

Abstract

An (upward) skip-free Markov chain with the set of nonnegative integers as state space is a chain for which upward jumps may be only of unit size; there is no restriction on downward jumps. In a 1987 paper, Brown and Shao determined, for an irreducible continuous-time skip-free chain and any d, the passage time distribution from state 0 to state d. When the nonzero eigenvalues nu_j of the generator are all real, their result states that the passage time is distributed as the sum of d independent exponential random variables with rates nu_j. We give another proof of their theorem. In the case of birth-and-death chains, our proof leads to an explicit representation of the passage time as a sum of independent exponential random variables. Diaconis and Miclo recently obtained the first such representation, but our construction is much simpler. We obtain similar (and new) results for a fastest strong stationary time T of an ergodic continuous-time skip-free chain with stochastically monotone time-reversal started in state 0, and we also obtain discrete-time analogs of all our results. In the paper's final section we present extensions of our results to more general chains.