Recently, Billard (1973) derived a solution to the forward equations of the general stochastic model. This solution contains some recursively defined constants. In this paper we solve these forward equations along each of the paths the process can follow to absorption. A convenient method of combining the solutions for the different paths results in a simplified non-recursive expression for the transition probabilities of the process.

Let {X1 (t), X2 (t), t ≧ 0} be a bivariate birth and death (Markov) process taking non-negative integer values, such that the process {X2(t), t ≧ 0} may influence the growth of the process {X1(t), t ≧ 0}, while the process X2 (·) itself grows without any influence whatsoever of the first process. The process X2 (·) is taken to be a simple linear birth and death process with λ2 and µ2 as its birth and death rates respectively. The process X1 (·) is also assumed to be a linear birth and death process but with its birth and death rates depending on X2 (·) in the following manner: λ (t) = λ1 (θ + X2 (t)); µ(t) = µ1 (θ + X2 (t)). Here λ i, µi and θ are all non-negative constants. By studying the process X1 (·), first conditionally given a realization of the process {X2 (t), t ≧ 0} and then by unconditioning it later on by taking expectation over the process {X2 (t), t ≧ 0} we obtain explicit solution for G in closed form. Again, it is shown that a proper limit distribution of X1 (t) always exists as t→∞, except only when both λ1 > µ1 and λ2 > µ2. Also, certain problems concerning moments of the process, regression of X1 (t) on X2 (t); time to extinction, and the duration of the interaction between the two processes, etc., are studied in some detail.

The multitype Galton-Watson process is considered both with and without immigration. Proofs are given for the existence of invariant measures and their uniqueness is examined by functional equation methods. Theorem 2.1 proves the uniqueness, under certain conditions, of solutions of a multidimensional Schröder equation. Regular variation is shown to play a role in the multitype theory.

Let w1, w12 and w2 be the fitnesses of genotypes A1A1, A1A2 and A2A2 in an infinite diploid population, and let pn be the A1, gene frequency in the nth generation. If fitness varies independently from generation to generation, then pn is a Markov process with a continuum of states. If E[In(wi/w12)] < 0 for i = 1, 2, then there is a unique stationary probability, and the distribution of pn converges to it as n → ∞.

I show that the sum of independent random variables converges in distribution when suitably normalised, so long as the Xk satisfy the following two conditions: μ(n)= E |Xn| is comparable with E |Sn| for large n, and Xk/μ(k) converges in distribution. Also I consider the associated birth process X(t) = max{n: Sn ≦ t} when each Xk is positive, and I show that there exists a continuous increasing function v(t) such that for some variable Y with specified distribution, and for almost all u. The function v, satisfies v (t) = A (1 + o (t)) log t. The Markovian birth process with parameters λn = λn, where 0 < λ < 1, is an example of such a process.

A Zipf's law is a probability distribution on the positive integers which decays algebraically. Such laws describe (approximately) a large class of phenomena. We formulate a model for such phenomena and, in terms of our model, give necessary and sufficient conditions for a Zipf's law to hold.

A selection of Bartlett's work as an operating statistician in his first position as statistician 1934–38 at the I.C.I. agricultural research station at Jealott's Hill, Berks., is described. This illustrates some of the methods he used for the efficient detection of treatment effects and for an appraisal of the suitability of the experimental designs that were being used.

This paper extends results of Esary, Marshall and Proschan (1973) and A-Hameed and Proschan (1973). We consider the life distribution of a device subject to a sequence of shocks occurring randomly in time according to a nonstationary pure birth process: given k shocks have occurred in [0, t], the probability of a shock occurring in (t, t + Δ] is λ kλ (t)Δ + o (Δ). We show that various fundamental classes of life distributions (such as those with increasing failure rate, or those with the ‘new better than used' property, etc.) are obtained under appropriate assumptions on λ k, λ (t), and on the probability of surviving a given number of shocks.

We investigate the large-sample behaviour of maximum likelihood estimates (MLE's) of the parameters of a diffusion process, which is observed throughout continuous time. The results (limit normal distribution for the MLE and an asymptotic chi-squared likelihood ratio test) correspond exactly to classical asymptotic likelihood results, and follow easily from a central limit theorem for stochastic integrals.

Reciprocal pairs of continuous random variables on the line are considered, such that the density function of each is, to within a norming factor, the characteristic function of the other. The analogous reciprocal relationship between a discrete distribution on the line and a continuous distribution on the circle is also considered. A conjecture is made regarding infinite divisibility properties of such pairs of random variables. It is shown that the von Mises distribution is infinitely divisible for sufficiently small values of the concentration parameter.

The literature on supply models for manpower planning in a hierarchy shows that an important consideration is the size of the discrepancy between the age distribution of the population and the stationary age distribution which would be reached if present policies were continued indefinitely. We establish the convergence properties of age distribution in various cases and show under what circumstances a convergent age distribution exists. In the convergent case we examine both the speed of convergence and the form of the stationary age distribution.

The general epidemic model does not provide for an infective recovering and thence being susceptible to further infection in the course of the epidemic. By considering the case in which recovery can occur once, we show how the state probabilities can be found for the stochastic model. This is readily extended to allow recovery up to a finite number of times.

A random collision process with transition probabilities belonging to the same type of distribution is considered. It was proved that if the characteristic function of the initial distribution has a positive radius of convergence, then the sequence {Fn(x)} converges weakly to a distribution G(x) [9]. We define a metric e1[Fn;G], which is analogous to the functional e[f] introduced by Tanaka to Kac's model of Maxwellian gas [10]. We prove that e1[Fn; G] is monotone non-increasing as n → ∞, and also the convergence of the sequence {Fn(x)} under the weaker assumption that for some a > 1 the initial distribution has an ath moment.

A class of point processes is considered, in which the locations of the points are independent random variables. In particular some properties of the process in which the distribution function of the position of the nth event is the n-fold convolution of some distribution function F, are investigated. It is shown that, under fairly general conditions, the process remote from the origin will be asymptotically Poisson. It is also shown that the variance of the number of events in the interval (0, t] is . Some generalisations are discussed.

In this article we consider a generalisation of conservative processes in which the usual transition rate parameters are replaced by time-dependent stochastic variables. The main result of the article shows that these generalised processes which we call conservative processes with stochastic rates have transition probabilities which can be characterised in terms of exchangeable random variables in a manner similar to the characterisation of conservative processes in terms of independent random variables given by Bartlett (1949). We use this characterisation to obtain general expressions for the transition probabilities and to examine some limiting aspects of the processes. The carrier-borne epidemic is treated as a particular case of these generalised processes.

The author determines the distribution and the limit distribution of the number of partial sums greater than k (k = 0, 1, 2, …) for n mutually independent and identically distributed discrete random variables taking on the integers 1, 0, − 1, − 2, ….

Bounds are derived for the probability of extinction by the nth generation for a branching process in a varying environment. From these bounds, necessary and sufficient conditions are established for such a process to become extinct with probability one. The extinction time of a random environment branching process in which the environmental random variables are independent but not necessarily identically distributed is stochastically bounded by the extinction times of two varying environment processes.

A discrete time, scalar, pursuit-evasion game is presented in which an evader, moving according to a stationary stochastic process, is continually being followed by a pursuer. Both players have perfect observations of the evader's positions, but the observations of the pursuer are subject to a time lag. It is assumed that the strategy of the evader can be represented as an infinite moving average, and that he is restricted by a velocity constraint. The pursuer is limited to strategies linear in his information, and the payoff is taken to be the mean-square distance between pursuer and evader. Under these conditions it is shown that the game does not have a value, and subsequently the lower and upper values and corresponding strategies are found.

The familiar problem of maximizing the probability of choosing the best from a group of N candidates, where N is known, is extended to the case of N unknown. An a priori distribution is assumed for N, and the case of a uniform distribution is examined. Let VN denote the probability of choosing the best from a group of at most N candidates, then it is shown that limN→∞VN = 2e–2.