We consider an important class of integer valued stochastic processes characterised by a certain form of time dependent probability generating function (P.G.F.). Certain results are obtained concerning the correlation structure of such processes, and it is shown that these are considerably simplified in the case of a stationary distribution having been attained. The results are illustrated in the latter case using some commonly occurring processes belonging to the class under consideration.

This paper discusses the response process when a Poisson process interacts with a renewal process in such a way that one or more points of the Poisson process eliminate a random number of consecutive points of the renewal process. A queuing situation is devised such that the c.d.f. of the length of the busy period is the same as the c.d.f. of the length of time intervals of the renewal response process. The Laplace-Stieltjes transform is obtained and from this the expectation of the time intervals of the response process is derived. For a special case necessary and sufficient conditions for the response process to be a Poisson process are found.

Several authors have considered the covariance structure of continuous parameter Markov chains. Most of this work has dealt with particular process ses, notably Morse (1955) who analysed the simple M/M/1 queue and Bene-(1961) who considered a telephone trunking model. Furthermore, the results obtained apply only when the process has attained its limiting (stationary) distribution. A recent paper by Reynolds (1968) gave some general results for finite chains, still assuming stationarity. This note generalises the results obtained therein, and considers the covariance structure during the transient period prior to attaining the stationary distribution where this exists. In the case where no such distribution exists, the results are valid throughout the whole lifetime of the process.

A necessary and sufficient condition based on order statistics that a positive random variable having an absolutely continuous probability distribution (with respect to Lebesgue measure) will be exponential is given.

A limit theorem for inter-record times is obtained which seems to explain many of their known asymptotic properties. The process of inter-record times is shown to be approximately non-homogeneous Poisson.

To obtain a limit with independent components in the superposition of m-dimensional point processes, a condition corresponding to asymptotic independence must be included. When this condition is relaxed, convergence to limits with dependent components is possible. In either case, convergence of finite distributions alone implies tightness and thus weak convergence in the function space D[0, ∞) × … × D[0, ∞).

A discrete time Markov chain with (n + 1) states is constructed in which there are n states having different decay parameters for geometric ergodicity.

A method, making use of characteristic functions, is presented to prove theorems on characterization of distributions by independence of order statistics. Since exact independence in practice is rarely achieved, much attention is being given in the literature to theorems on determining the distributions for which certain statistics are ‘almost’ independent. The technique of this new line of investigation is strongly dependent on the fact that for the case of exact independence there is a solution in terms of characteristic functions. The present work was guided by this fact. In addition to the new method, Theorem 2 appears to be new.

In a single-shelf library of N books we suppose that books are returned by placing them at one end of the shelf. The probability of selecting each book is assumed to be known, and only one book is removed at a time. The N! arrangements of the books are considered as states of an ergodic Markov chain for which we find the stationary distribution.