Regenerative Phenomena and the Characterization of Markov Transition Probabilities

It is the object of this paper to draw together certain lines of research which during the last decade have grown out of the problem of characterizing the functions which can arise as transition probabilities of continuous time Markov chains. This problem is now solved (see Sections 8 and 9). although as usual its solution has thrown up further problems which demand attention. The evolution of a Markov chain X, in continuous time, with stationary transition probabilities, on a countable state space S. is as usual [2] described by the functions