Solidarity and ergodic properties of semi-Markov transition probabilities

Abstract

The chief purpose of this thesis is to establish for semi-Markov processes the same type of behaviour that is characteristic of the better-known Markov chains; this is achieved mainly through the use of Laplace and Laplace-Stieltjes transforms and a frequent appeal to renewal theory. The mathematical tools needed for the task are developed in the first chapter. In the second chapter the solidarity nature of geometric ergodicity within an irreducible class is examined, and necessary and sufficient conditions are derived for geometric ergodicity in the particular case of a process with a finite state space. In chapter three it is shown that the Laplace transforms of the transition probabilities pertaining to an irreducible class all have the same abscissa of convergence, a fact that permits the definition of a-recurrence and leads to a result for a-recurrent processes that generalizes the familiar ergodic theorem of Markov chain theory; quasi-stationary distributions are also studied in the same chapter. Chapter four is devoted to some general ratio-limit theorems involving a parameter λ (λ equals zero in the usual ratio-limit theorems), and the last chapter applies the results obtained in the earlier part of the thesis to the study of an inventory model and a continuous-time Markov branching process.