Quasi-limiting distributions of Markov chains that are skip-free to the left in continuous time

Abstract

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.