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.

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 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.

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.