Non-recurrent stationary stochastic point processes.

The work is mainly concerned with the general theory of stationary point processes and the theory of some particular stationary point processes. The intervals separating events are dealt with by the introduction of 'average events' and 'arbitrary events'. An average event is based on the average of the first n events as n tends to infinity; an arbitrary event is based on the notion of an instant of at least one event. The latter leads to modifications and extensions of some work contained in Khintchine (1960, Mathematical Methods in the Theory of Queuing). A general theory of stationary point processes is built up in which the assumptions for the results of McFadden (1962, J.R.Statist.Soc.B.,28) are clarified. A relation connecting the arbitrary and basic random variables is obtained which does not depend on the events occurring distinctly or the arbitrary intervals forming a stationary sequence (Wold stationarity). Four particular point processes are then discussed in detail. The pooling of point process, in the sense of Cox and Smith (1954, Biometrika, 41), is considered both by the average and arbitrary event approaches. The joint distributions of up to four average intervals are obtained for the pooling of two renewal processes, and Wold stationarity is verified. The pooling of any number of general stationary point processes is then dealt with by the arbitrary event approach. Next the 'renewal inhibited Poisson process' of Ten Hoopen and Reuver (l965, J.Appl.Prob.,2) is treated as a point process. The joint average interval distribution, indicating its Wold stationarity, is obtained and the counting processes of events, both in the stationary and synchronous cases, are derived. A joint process covering all aspects of the process is investigated. The work on special process is completed with the stationary point theories of semi-Markov processes and the 'random hazard process' of Gaver (1963, Technometrics, 5). Computer simulations and extensions of the processes are discussed.