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.

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.

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.

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.

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 M 1 topology. Selected applications are discussed, one being to the Gini coefficient.

This two-part paper surveys some recent developments in integral and stochastic geometry. Part I surveys applications of integral geometry to the theory of euclidean motion-invariant random fibrefields (a fibrefield is a collection of smooth arcs on the plane), involving marked point processes, Palm distribution theory and vertex pattern analysis. Part II develops the more sophisticated theory of Buffon sets in stochastic geometry and the characterisation of measures of lines, giving applications to problems concerning random triangles and colourings, line processes and fixed convex sets.

Recent research on topics related to geometrical probability is reviewed. The survey includes articles on random points, lines, line-segments and flats in Euclidean spaces, the random division of space, coverage, packing, random sets, stereology and probabilistic aspects of integral geometry.