Screening for an early detection of disease, particularly when the screening tests have to be carried out on a number of occasions during the lifetime of a person, is a complex issue and the development of suitable models that can help in setting up good screening programmes for a community is a challenging task. In most cases the current emphasis is on screening at regular intervals of time and the frequency of screening is determined, usually in an informal way, by factors like the nature of the test and the nature of the disease. Analysis of a simple two-state model leads to a basic building-block which can be used in the construction of models for a variety of screening programmes. The use of this building block is illustrated and it is shown that the determination of non-regular screening times in a formal manner can lead to a greater efficiency.

Grade and age structures in manpower systems are often far from ideal. This fact raises the question of how the flows of people — and particularly the recruitment flow — should be controlled in order to attain and maintain a more desirable structure. The problem has received considerable attention from a deterministic point of view. This paper adopts a stochastic approach to the study of maintainability and shows, among other things, that the problem is more subtle than the deterministic analysis suggests.

The age structure of a large, unisexual, closed population is described here by a vector of the proportions in each age class. Non-negative matrices of age-specific birth and death rates, called Leslie matrices, map the age structure at one point in discrete time into the age structure at the next. If the sequence of Leslie matrices applied to a population is a sample path of an ergodic Markov chain, then: (i) the joint process consisting of the age structure vector and the Leslie matrix which produced that age structure is a Markov chain with explicit transition function; (ii) the joint distribution of age structure and Leslie matrix becomes independent of initial age structure and of the initial distribution of the Leslie matrix after a long time; (iii) when the Markov chain governing the Leslie matrix is homogeneous, the joint distribution in (ii) approaches a limit which may be easily calculated as the solution of a renewal equation. A numerical example will be given in Cohen (1977).

The paper deals with the asymptotic behaviour of infinite mean Galton–Watson processes (denoted by {Zn}). We show that these processes can be classified as regular or irregular. The regular ones are characterized by the property that for any sequence of positive constants {Cn}, for which a.s. exists, The irregular ones, which will be shown by examples to exist, have the property that there exists a sequence of constants {Cn} such that In Part 1 we study the properties of {Zn/Cn} and give some characterizations for both regular and irregular processes. Part 2 starts with an a.s. convergence result for {yn(Zn)}, where {yn} is a suitable chosen sequence of functions related to {Zn}. Using this, we then derive necessary and sufficient conditions for the a.s. convergence of {U(Zn)/Cn}, where U is a slowly varying function. The distribution function of the limit is shown to satisfy a Poincaré functional equation. Finally we show that for every process {Zn} it is possible to construct explicitly functions U, such that U(Zn)/en converges a.s. to a non-degenerate proper random variable. If the process is regular, all these functions U are slowly varying. The distribution of the limit depends on U, and we show that by appropriate choice of U we may get a limit distribution which has a positive and continuous density or is continuous but not absolutely continuous or even has no probability mass on certain intervals. This situation contrasts strongly with the finite mean case.

We consider several problems in the theory of first-passage percolation on the two-dimensional integer lattice. Our results include: (i) a mean ergodic theorem for the first-passage time from (0,0) to the line x = n; (ii) a proof that the time constant is zero when the atom at zero of the underlying distribution exceeds C, the critical percolation probability for the square lattice; (iii) a proof of the a.s. existence of routes for the unrestricted first-passage processes; (iv) a.s. and mean ergodic theorems for a class of reach processes; (v) continuity results for the time constant as a functional of the underlying distribution.

The testing of the hypothesis that a point process is Poisson against a one-dimensional alternative is considered. The locally optimal test statistic is expressed as an infinite series of uncorrelated terms. These terms are shown to be asymptotically equivalent to terms based on the various orders of cumulant spectra. The efficiency of tests based on partial sums of these terms is found.

The problem of scheduling items for service with random service times is formulated as an optimal control problem. Pontryagin's maximum principle is used to determine the optimal schedule in certain cases.

In a wide variety of multi-wave panel studies in economics and sociology, comparisons between the observed transition matrices and predictions of them based on time-homogeneous Markov chains have revealed a special kind of discrepancy: the trace of the observed matrices tends to be larger than the trace of the predicted matrices. A common explanation for this discrepancy has been via mixtures of Markov chains.

Specializing to mixtures of Markov semi-groups of the form

we exhibit classes of stochastic matrices M, probability measures µ and time intervals Δ such that for k = 2, 3 and 4. These examples contradict the substantial literature which suggests — implicitly — that the above inequality should be reversed for general mixtures of Markov semi-groups.

An occupation measure describes the expected amount of time a stochastic process spends in different parts of its state space prior to a given random time. It is shown that a basic identity involving occupation measures provides a unified approach to a variety of moment identities for Markov chains, and some connections with potential theory are made.

The Lorenz curve of the distribution of ‘wealth’ is a graph of cumulative proportion of total ‘wealth’ owned, against cumulative proportion of the population owning it. This paper uses Gastwirth's definition of the Lorenz curve which applies to a general probability distribution on (0, ∞) having finite mean; thus it applies both to a ‘population’ distribution and to empirical distributions obtained on sampling. The Lorenz curves of the latter are proved to converge, with probability 1, uniformly to the former, and similarly for their inverses. Modified Lorenz curves are also defined, which treat atoms of differently, and these and their inverses are proved strongly consistent. Functional central limit theorems are then proved for empirical Lorenz curves and their inverses, under condition that be continuous and have finite variance. A mild variation condition is also needed in some circumstances. If the support of is connected, the weak convergence is relative to C[0, 1] with uniform topology, otherwise to D[0, 1] with M1 topology. Selected applications are discussed, one being to the Gini coefficient.