Approximate results are obtained for the probability distribution of the number of mutant substitutions over a given number of generations. The distribution is a compound Poisson distribution having a variance which is greater than the mean. The substitutions occur in clusters. The most extreme case of clustering is studied in this paper, which assumes that there is no recombination between mutant nucleotides.

A study is made of the joint distribution of the number of alleles and segregating sites in a random sample of genes from the infinite-alleles model of population genetics. Tables of the joint distribution are given for representative values of the mutation parameter. A conclusion is that each segregating site has a high probability of delineating an allele type.

The purpose of this paper is to study simple mathematical models for the variation of the concentration of atmospheric pollutants. The particles are assumed to be emitted into the atmosphere with constant intensity and removed from the atmosphere with an intensity proportional to the precipitation intensity. The precipitation intensity is assumed to be a stationary stochastic process with no spatial variation. The mean, variance and covariance function of the concentration are calculated for some precipitation models. Under certain general assumptions it is shown that the concentration of ‘long-lived' particles is approximately described by an Ornstein–Uhlenbeck process. Finally, the different models are compared by means of Swedish precipitation data.

The stochastic process {Xn} satisfying Xn+1 = max{Yn+1+ αβ Xn, βXn} where {Yn} is a stationary sequence of non-negative random variables and , 0<β <1, can be regarded as a simple thermal energy storage model with controlled input. Attention is mostly confined to the study of μ = EX where the random variable X has the stationary distribution for {Xn}. Even for special cases such as i.i.d. Yn or α = 0, little explicit information appears to be available on the distribution of X or μ . Accordingly, bounding techniques that have been exploited in queueing theory are used to study μ . The various bounds are illustrated numerically in a range of special cases.

After an elementary description of Gelfand pairs, spherical functions and Plancherel measure, some explicit computations on the related Markov chains are performed. Random walks on polyhedra belong to this class of Markov chains; two more examples of chains on graphs are worked out, and the necessary and sufficient condition of transcience of random walks on p-adic numbers with spherical symmetry is given as an application of the techniques of the paper.

A class of two-dimensional birth-and-death processes, with applications in many modelling problems, is defined and analysed in the steady state. These are processes whose instantaneous transition rates are state-dependent in a restricted way. Generating functions for the steady-state distribution are obtained by solving a functional equation in two variables. That solution method lends itself readily to numerical implementation. Some aspects of the numerical solution are discussed, using a particular model as an example.

Let be an optional stochastic process and let be the directed set of almost-surely-finite stopping times. If lim , then

This equality is employed in studying the continuous-time gambler's problem, as formulated by Heath and Sudderth (1974). The optimal return function is shown to be upper semi-analytic and excessive whenever the utility function is Borel.

Let Yn denote the largest of n independent N(0,1) random variables. It is shown that the error in approximating the distribution of Yn by the type III extreme value distribution exp {– (–Ax + B)k}, k > 0, is uniformly of order (log n)–2 if and only if the constants A, B and k satisfy certain conditions. In particular, this holds for the penultimate form of Fisher and Tippett (1928). Furthermore, two sufficient conditions are given so that these results can be extended to a stationary Gaussian sequence.

n i.i.d. random variables with known continuous distribution function F are observed sequentially with the object of choosing the largest. After any observation, say the kth, the observer may solicit any of the first k observations. If the (k – t)th is solicited, the probability of a successful solicitation may depend on t, the number of observations since the (k – t)th, and on the quantile of the (k – t)th observation. General properties of optimal selection procedures are obtained and the optimal procedures and their probabilities of success are derived in some special cases.