Some passage-time generating functions for discrete-time and continuous-time finite Markov chains

Abstract

Let T denote a subset of the possible transitions between the states of a finite Markov chain and let Y_k denote the time of the kth occurrence of a T-transition. Formulae are derived for the generating functions of Y_k, of Y_j + k — Y_j and of Yj + k — Y_j in the limit as j → ∞, for both discrete-time and continuoustime chains. Several particular cases are briefly discussed.