In this paper is developed a theory of branching processes in which the division probability and rate of growth of cells depend only on their ‘size’ and in which ‘size’ is shared between daughters. Specific results are obtained in a linear case including the calculation of the correlation coefficient for the life spans of sisters.

In a multi-type critical age-dependent branching process with immigration, the numbers of cell types alive at time t, each divided by t, as t becomes large, tends to a one-dimensional gamma distribution law. The method of proof employs generating functions and compution of asymptotic moments. Connections with earlier results and extensions are indicated.

In the Bellman-Harris (B-H) age-dependent branching process, the birth of a child can occur only at the time of its parent's death. A general class of branching process in which births can occur throughout the lifetime of a parent has been introduced by Crump and Mode. This class shares with the B-H process the property that the generation sizes {ξn} form a Galton-Watson process, and so may be classified into subcritical, critical or supercritical according to the value of m = E{ξ1}. Crump and Mode showed that, as regards extinction probability, asymptotic behaviour, and for the supercritical case, convergence in mean square of Z(t)/E[Z(t)], as t → ∞, where Z(t) is the population size at time t given one ancestor at t = 0, properties of the B-H process can be extended to this general class. In this paper conditions are found for the convergence in distribution of Z(t)/E{Z(t)} in the supercritical case to a non-degenerate limit distribution. In contrast to the B-H process, these conditions are not the same as those for ξn/mn to have a non-degenerate limit. An integral equation is established for the generating function of Z(t), which is more complicated than the corresponding one for the B-H process and involves the conditional probability generating functional of N(x), x 0, ≧ the number of children born to an individual in the age interval [0, x].

The general minimum mean square error prediction problem is studied for the class of processes defined by polynomial functions of the Ornstein-Uhlenbeck process. In particular, the relative error difference for non-linear prediction compared with optimal linear prediction is calculated. Limiting values are determined, including an upper bound for the maximal relative error difference. These results show no difference over the entire class for both long and short prediction lead times. For a single Hermite polynomial, the relative error difference is zero. An analytical solution for the maximal difference for lead times below a given threshold is derived. This latter result has been verified numerically on a digital computer for polynomials up to 17th degree.

A random process on the circle is a family of random variables X(P,t) indexed by the position P on the unit circle and by the time t (t = 0, + 1, ···). For a homogeneous and stationary process, using its Fourier series expansion, we deduce a spectral representation of the covariance function. The purpose of the paper is to develop a spectral analysis when X(P,t) is observed at all the points on the circle at t = 0, 1, ···, T – 1. The asymptotic distribution of the family of finite Fourier transforms, of the family of periodograms and of the family of spectral densities estimates are obtained from results available for vector-valued time series. Also, a sample covariance function for X (P,t) is defined and its asymptotic distribution derived.

A general multivariate non-linear regression model is considered, including as special cases linear regression when the regression matrix is of less than full rank, simultaneous equations systems and regression on an unobservable predetermined variable. Given a time-series of observations at unit intervals we consider the estimation of the parameters, subject to non-linear constraints, by minimizing a criterion based on the Fourier-transformed model. We allow the residuals to be generated by a stationary, linear, process and establish asymptotic properties of our estimates.

A result in Stone, Belkin, and Snyder ((1970) J. Math. Anal. Appl.30, 448–470) gave a method for finding the Laplace-Stieltjes transform of the distribution of certain non-negative, homogeneous, additive functionals of a Markov process with stationary transition measure. By considering certain two dimensional Markov processes and applying this result, a method is obtained for finding time above a threshold and first passage distributions for a one dimensional process either when (1) the process is Markovian and the threshold is possibly non-constant, or (2) the threshold is constant and the process is the indefinite integral of a Markov process. Specific process-threshold combinations are considered in several examples including the case of a linear threshold for a Wiener process and a for compound Poisson process with exponential (either one-sided or bilateral) after-jump distribution. In addition, the first passage distribution to a constant threshold is computed for an integrated Poisson sampling process.

Let {Zt, t ≧ 0} be an irreducible regular semi-Markov process with transition probabilities Pij (t). Let f(t) be non-negative and non-decreasing to infinity, and let λ ≧ 0. This paper identifies a large set of functions f(t) with the solidarity property that convergence of the integral ≧ eλtf(t)Pij(t) dt for a specific pair of states i and j implies convergence of the integral for all pairs of states. Similar results are derived for the Markov renewal functions Mij (t). Among others it is shown that f(t) can be taken regularly varying.

Jensen gave a lower bound to Eρ(T), where ρ is a convex function of the random vector T. Madansky has obtained an upper bound via the theory of moment spaces of multivariate distributions. In particular, Madansky's upper bound is given explicitly when the components of T are independent random variables. For this case, lower and upper bounds are obtained in the paper, which uses additional information on T rather than its mean (mainly its expected absolute deviation about the mean) and hence gets closer to Eρ(T).

The importance of having improved bounds is illustrated through a nonlinear programming problem with stochastic objective function, known as the “wait and see” problem.

“Recursive” games were first defined and studied by Everett. Related results can be found in Gillette, Milnor and Shapley, and Blackwell and Ferguson. In this paper we introduce the notion of a recursive matrix game, which we believe eliminates the vagueness but none of the useful generality of the earlier definition. We then give an inductive proof (different from the proof in [3]) that these games have a value, with ∊-optimal stationary strategies available to each player. We also apply the result and show how a class of games studied in a different framework are games of this type and thus have a value.

This paper demonstrates that, when in heavy traffic, the quasi-stationary distribution of the virtual waiting time process of both the M/G/1 and GI/M/1 queues as well as the quasi-stationary distribution of the waiting times {Wn} of the M/G/1 queue can be approximated by the same gamma distribution. What characterises this approximating gamma distribution are the first two moments of the service time and inter-arrival time distributions only. A similar approximating behaviour is demonstrated for the queue size process.

A general procedure is outlined for obtaining lower bounds and approximations to the amount of congestion in queues with low traffic. Some detailed formulae are given for a number of single-server systems and compared with exact solutions where available. Results are also given for a discrete time system in which departures clash with new arrivals.

Balls are drawn with replacement from an urn containing m distinguishable balls until a match is noted. The distribution of the number of drawings required is considered in the case where only the last k balls drawn are remembered. The asymptotic behavior of this distribution, as m and k become large, is investigated. Two further variants of the problem are suggested.

Limit theorems for the thinning of renewal point processes according to two different schemes are studied. In the first scheme when a point is retained a random number of succeeding points are deleted. According to the second scheme a random number of points are deleted by an inhibitory Poisson process.

The conjecture pronounced at the end of the paper of Srivastava and Srivastava (1970) is proved in this paper. It gives the following characterization of (bivariate) Poisson distributions. Suppose that items of two types have been observed certain numbers of times, but these original observations have been reduced due to a destructive process which is the product of two binomial distributions and that the probabilities of these reduced numbers are the same whether damaged or undamaged. Then the original random variables had a bivariate Poisson distribution with zero mutual dependence coefficient.

In his paper ‘Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test’ (J. Appl. Prob.8, 431–453), Durbin derives an integral equation whose kernel has a singularity. Since direct solution of an approximating set of simultaneous equations would be very inaccurate, he uses probability arguments to approximate to integrals of sub-intervals. In this note, two alternative procedures are discussed. One makes a linear transformation of the original integral equation to eliminate the singularity; the other, due to Weiss and Anderssen, integrates the singular factor in the kernel over the sub-interval. Computation of a special case indicates this latter method to be the most effective of the three.

This paper aims at showing that for the discrete time analogue of the M/G/l queueing model with service in random order and with a traffic intensity ρ > 0, the condition ρ < ∞ is sufficient in order that every customer joining the queue be served eventually, with probability one (Theorem 2).

It is proved that in a GI/G/s queue with the traffic intensity ρ < (S – m + l)/S, m ≧ 1, at least m servers are simultaneously idle infinitely often with probability one.

In a recent paper (Downton (1972)) an attempt was made to use the combinatorial approach of Downton (1967) to obtain an expression for the Laplace transform (and hence the moments) of the area under the infectives trajectory of the general stochastic epidemic. The basic property of that area, which makes this possible, is that it is composed of a random number of rectangles, each of whose area has a distribution depending on the infection rate and the number of susceptibles present at the appropriate stage in the development of the epidemic, but which is independent of the number of infectives. However, Mr A. Abakuks has pointed out that in one respect the argument used was faulty. It was argued that the contribution to the area under the infectives curve of an interval ending in a reduction in the number of infectives from i to i – 1 had an exponential distribution with parameter p (the infection rate); and if the interval ended in a reduction in the number of susceptibles from s to s – 1 the distribution was exponential with parameter s. While this is true unconditionally, the combinatorial situation described was essentially a conditional one, in which each of these contributions to the total area had an exponential distribution, independent of i, but with parameter (ρ + s) regardless of whether the interval ended in a reduction in the number of infectives or of susceptibles.