Consider an open multiclass Jacksonian network in equilibrium and a path such that a customer travelling along it cannot be overtaken directly by a subsequent arrival or by the effects of subsequent arrivals. Then the sojourn times of this customer in the nodes constituting the path are all mutually independent and so the total sojourn time is easily calculated. Two examples are given to suggest that the non-overtaking condition may be necessary to ensure independence when there is a single customer class.

For family data where some of the children may have a certain inherited disorder, the number of affected children has usually been assumed to be binomially distributed given the total number of children in the family. In this assumption is included the assumption that the distribution of the total number of children does not depend on the probability that a child is affected. For many disorders this assumption is unlikely to hold because the birth of an affected child may lead either to some sort of family limitation or to some sort of overcompensation. In such cases models other than the binomial distribution have to be used.

Many medical genetics studies are at least partially sequential, and in considering problems of ascertainment it seems desirable to consider such schemes, in addition to classical ascertainment models. Cannings and Thompson (1978) indicated the correction to the likelihood required when sampling a pedigree sequentially from some predetermined initial event, and Thompson and Cannings (1979) have extended this result and compared sequential sampling with more classical ascertainment models.

Fischer (1930) stated his fundamental theorem of natural selection (ftns) as ‘the rate of increase in fitness of any organism at any time is equal to its genetic (additive) variance in fitness at that time’. The theorem related to an idealised population and he thought that it held ‘the supreme position among the biological sciences’. His proof of the theorem is, however, not understood. This has given rise to questions concerning not only the validity of the proof but also concerning the importance of the theorem.

For N = 1, 2, ···, let {(XN(k), YN(k)), k = 0, 1, ···} be a homogeneous Markov chain in ℝm x ℝn. Suppose that, asymptotically as N → ∞, the ‘infinitesimal’ covariances and means of XN([·/∊N]) are aij(x, y) and bi(x, y), and those of YN([·/δN]) are 0 and cl(x, y). Assume limN→∞ δN = limN→∞ ∊N/δN = 0 and the zero solution of ý = c(x, y) is globally asymptotically stable. Then, under some technical conditions, it is shown that (i) XN([·/∊N]) converges weakly to a diffusion process with coefficients aij(x, 0) and bi(x, 0), and (ii) YN([t/∊N]) → 0 in probability for every t > 0. (The case limN→∞ δN = δ∞ > 0 = limN→∞ ∊N is also treated.) The proof is based on the discrete-parameter analogue of a generalization of Kurtz's limit theorems for perturbed operator semigroups.

The results are applied to three classes of stochastic models for random genetic drift at a multiallelic locus in a finite diploid population. The three classes involve multinomial sampling, overlapping generations, and general progeny distributions. Within each class, the monoecious, dioecious autosomal, and X-linked cases are analyzed. It is found that results for a monoecious population obtained from a diffusion approximation can be applied at once to the dioecious cases by using the appropriate effective population size and averaging allelic frequencies, selection intensities, and mutation rates, weighting each sex by the number of genes carried by an individual at the locus under consideration.

The ray solution of Felsenstein's n-allele random selection diffusion process is given for small values of the selection parameters. This solution holds away from, but not near, the boundary of frequency space. The solution is possible only because the coefficients of the associated Jacobi field equations agree uniformly with those for the case of zero selection up to fourth powers in the selection parameters, whilst the covariance of the diffusion has only quadratic dependence.

A birth–death process {x(t): t ≥ 0} with state space the set of non-negative integers is said to be stochastically increasing (decreasing) on the interval (t1, t2) if Pr {x(t) > i} is increasing (decreasing) with t on (t1, t2) for all i = 0, 1, 2, ···. We study the problem of finding a necessary and sufficient condition for a birth–death process with general initial state probabilities to be stochastically monotone on an interval. Concrete results are obtained when the initial distribution vector of the process is a unit vector. Fundamental in the analysis, and of independent interest, is the concept of dual birth–death processes.