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

James Allen Fill ()
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James Allen Fill: The Johns Hopkins University

Journal of Theoretical Probability, 2009, vol. 22, issue 3, 587-600

Abstract: 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 ν j 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 ν 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.

Keywords: Markov chains; Skip-free chains; Birth-and-death chains; Passage time; Absorption time; Strong stationary duality; Fastest strong stationary times; Eigenvalues; Stochastic monotonicity; 60J25; 60J35; 60J10; 60G40 (search for similar items in EconPapers)
Date: 2009
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Citations: View citations in EconPapers (11)

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DOI: 10.1007/s10959-009-0233-7

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