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1 results

  • Spectral structure of the first-passage-time densities for classes of Markov chains

  • Masaaki Kijima
  • Journal:
    Journal of Applied Probability / Volume 24 / Issue 3 / September 1987
    Published online by Cambridge University Press:
    14 July 2016, pp. 631-643
    Print publication:
    September 1987

  • Keilson [7] showed that for a birth-death process defined on non-negative integers with reflecting barrier at 0 the first-passage-time density from 0 to N (N to N + 1) has Pólya frequency of order infinity (is completely monotone). Brown and Chaganty [3] and Assaf et al. [1] studied the first-passage-time distribution for classes of discrete-time Markov chains and then produced the essentially same results as these through a uniformization. This paper addresses itself to an extension of Keilson's results to classes of Markov chains such as time-reversible Markov chains, skip-free Markov chains and birth-death processes with absorbing states. The extensions are due to the spectral representations of the infinitesimal generators governing these Markov chains. Explicit densities for those first-passage times are also given.