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.
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Special note: The editorial review of this article, and that of DOI 10.1007/s10959-009-0235-5, was managed by former Editor-in-Chief Arunava Mukherjea, who acted in the role of Editor-in-Chief. The two articles were submitted to Journal of Theoretical Probability in order to form a natural three-article sequence with DOI 10.1007/s10959-009-0234-6 (by Diaconis and Miclo).
Research supported by NSF grant DMS–0406104, and by The Johns Hopkins University’s Acheson J. Duncan Fund for the Advancement of Research in Statistics.
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Fill, J.A. On Hitting Times and Fastest Strong Stationary Times for Skip-Free and More General Chains. J Theor Probab 22, 587–600 (2009). https://doi.org/10.1007/s10959-009-0233-7
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DOI: https://doi.org/10.1007/s10959-009-0233-7
Keywords
- Markov chains
- Skip-free chains
- Birth-and-death chains
- Passage time
- Absorption time
- Strong stationary duality
- Fastest strong stationary times
- Eigenvalues
- Stochastic monotonicity