On Hitting Times and Fastest Strong Stationary Times for Skip-Free and More General Chains

By James Allen Fill. 2007

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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 νj of the generator on {0,…,d}, with d made absorbing, are all real, their result states that the passage time is distributed as the sum of d independent exponential random variables with rates νj. 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.

Subfields: Management Information Systems, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3e705ce36b13407f479f018731f200be20a95654

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Citations in the corpus

The Passage Time Distribution for a Birth-and-Death Chain: Strong Stationary Duality Gives a First Stochastic Proof (2007, background): “In a forthcoming paper [9], we will exhibit a much simpler such construction, with extensions to skip-free processes.”
On Times to Quasi-stationarity for Birth and Death Processes (2009, background): “In two recent papers [10, 11], Fill gives another proof of Proposition 1, but it relies on a nontrivial linear algebra result of Micchelli and Willoughby [20].”