A continuous-time Markov chain on the non-negative integers is called skip-free to the left (right) if the governing infinitesimal generator A = (aij) has the property that aij = 0 for j ≦ i ‒ 2 (i ≦ j – 2). If a Markov chain is skip-free both to the left and to the right, it is called a birth-death process. Quasi-limiting distributions of birth–death processes have been studied in detail in their own right and from the standpoint of finite approximations. In this paper, we generalize, to some extent, results for birth-death processes to Markov chains that are skip-free to the left in continuous time. In particular the decay parameter of skip-free Markov chains is shown to have a similar representation to the birth-death case and a result on convergence of finite quasi-limiting distributions is obtained.

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.