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Hitting time distribution for skip-free Markov chains: A simple proof

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  • Zhou, Ke

Abstract

A well-known theorem for an irreducible skip-free Markov chain on the nonnegative integers with absorbing state d, under some conditions, is that the hitting (absorbing) time of state d starting from state 0 is distributed as the sum of d independent geometric (or exponential) random variables. The purpose of this paper is to present a direct and simple proof of the theorem in the cases of both discrete and continuous time skip-free Markov chains. Our proof is to calculate directly the generation functions (or Laplace transforms) of hitting times in terms of the iteration method.

Suggested Citation

  • Zhou, Ke, 2013. "Hitting time distribution for skip-free Markov chains: A simple proof," Statistics & Probability Letters, Elsevier, vol. 83(7), pages 1782-1786.
    • Handle: RePEc:eee:stapro:v:83:y:2013:i:7:p:1782-1786
    DOI: 10.1016/j.spl.2013.04.008
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    Cited by:

    1. Emilio De Santis & Fabio Spizzichino, 2016. "Some Sufficient Conditions for Stochastic Comparisons Between Hitting Times for Skip-free Markov Chains," Methodology and Computing in Applied Probability, Springer, vol. 18(4), pages 1021-1034, December.
    2. Xiang, Xuyan & Fu, Haiqin & Zhou, Jieming & Deng, Yingchun & Yang, Xiangqun, 2021. "Taboo rate and hitting time distribution of continuous-time reversible Markov chains," Statistics & Probability Letters, Elsevier, vol. 169(C).

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