The class of fractional linear generating functions, one of the few known classes of probability generating functions whose iterates can be explicitly stated, is examined. The method of bounding a probability generating function g (satisfying g″(1) < ∞) by two fractional linear generating functions is used to derive bounds for the extinction time distribution of the Galton-Watson branching process with offspring probability distribution represented by g. For the special case of the Poisson probability generating function, the best possible bounding fractional linear generating functions are obtained, and the bounds for the expected time to extinction of the corresponding Poisson branching process are better than any previously published.

In a branching process with variable lifetime, introduced by Fildes (1972) define Yjk(t) as the number of particles alive in generation k at time t when the initial particle is born in generation j. A limit law similar to that derived in the Bellman-Harris process is proved where it is shown that Yjk(t) suitably normalised converges in mean square to a random variable which is the limit random variable of Znm–n in the Galton-Watson process (m is the mean number of particles born).

Haploid models of genetic drift in populations of constant size are considered. Generalizations of the models of Moran and Wright have been developed by Karlin and McGregor (for multiple alleles and non-overlapping generations), by Chia and Watterson (for two alleles and overlapping or non-overlapping generations) and by Chia (for multiple alleles and overlapping or non-overlapping generations), using conditioned branching processes. A new approach is developed which contains the models mentioned above and provides simpler expressions for the latent roots. A greater dependence between the birth events and death events can be permitted, and non-independent mutations treated.

There is now a large literature on linkage disequilibrium between pairs of loci, both for selection in infinite populations and for neutral genes in finite populations, but there have been few studies with more loci. Bennett (1954) showed how the frequencies of chromosomes with any number of neutral genes would change in an infinite population, and the author (unpublished) has extended Bennett's results to find expected changes in chromosome frequencies with up to six loci in finite populations. For two linked neutral genes in finite populations the expected disequilibrium is zero, but the variance of the disequilibrium or the correlation of gene frequencies in segregating populations has been found. This has been done by Monte Carlo simulation (Hill and Robertson (1968)), but an approximation can be obtained by diffusion methods (Ohta and Kimura (1969)) and the asymptotic values using inbreeding theory (Sved (1971)). In this note we discuss the case of disequilibrium between three neutral loci and show how it relates to that between two loci.

The following properties of entropies, as measures of expected information, seem natural. The amount of information expected from an experiment does not change if we add outcomes of zero probability (expansibility). The expected information is symmetric in the (probabilities of the) outcomes. The information expected from a combination of two experiments is less than or equal to the sum of the informations expected from the single experiments (subadditivity); equality holds here if the two experiments are independent (additivity).

In this paper it is shown that linear combinations of the Shannon and Hartley entropies and only these have the above properties. The Shannon and the Hartley entropies are also individually characterized.

Let {XN(t)} be a sequence of continuous time Markov population processes on an n-dimensional integer lattice, such that XN has initial state Nx(0) and has a finite number of possible transitions J from any state X: let the transition X → X + J have rate NgJ(N–1X), and let gJ(x) and x (0) be fixed as N varies. The rate of convergence of √N(N–1XN(t) — ζ (t)) to a Gaussian diffusion is investigated, where ζ(t) is the deterministic approximation to N–1XN(t), and a method of deriving higher order asymptotic expansions for its distribution is justified. The methods are applied to two birth and death processes, and to the closed stochastic epidemic.

Recent research on topics related to geometrical probability is reviewed. The survey includes articles on random points, lines, flats, and networks in Euclidean spaces, pattern recognition, random coverage and packing, random search, stereology, and probabilistic aspects of integral geometry.

The imago of the peppered moth exists in two sharply distinguished forms, speckled and black. Clarke and Sheppard (1966) have shown that the frequency of the black moth decreases from 99% in Liverpool to 10% in North Wales. If these frequencies are in equilibrium, albeit dynamic equilibrium, then such a gradient is called a cline. Haldane (1948) briefly describes several clines, and then defines a model that may represent the life-cycle of certain species. The assumptions defining his model are as follows.

A classical question in population genetics is ‘In a species in which there is obligate outcrossing is there an increased amount of heterozygosity?’. This somewhat vague question can have two different meanings: (1) Is the number of alleles maintained at a locus greater? and (2) If there are two alleles at a particular locus is the number of heterozygotes formed greater? It is the last question which is discussed here.

In this paper we consider two optimization problems and two game problems. In each problem, a particle is hidden on the real line (sometimes randomly, and sometimes by an antagonistic hider), and a seeker, starting at the origin, wishes to find the particle with minimal expected cost. We consider a fairly wide class of cost functions depending upon the position of the particle and the time used to discover it. For the games we obtain the values and (∊-) optimal strategies. For the optimization problems we obtain qualitative features of (∊-) optimal searches.

We consider a finite controlled Markov chain, the description of which depends on an unknown parameter a, and investigate the following control policy. To each a an optimal stationary control is associated. a is estimated recurrently from the trajectory by the minimum contrast method, and the optimal stationary control corresponding to the estimate is used. We present asymptotic properties of the estimate and of the criterion function. They follow from the law of large numbers and from the central limit theorem for controlled Markov chains derived with the aid of martingales.