This paper considers certain stochastic control problems in which control affects the criterion through the process trajectory. Special analytical methods are developed to solve such problems for certain dynamical systems forced by white noise. It is found that some control problems hitherto approachable only through laborious numerical treatment of the non-linear Bellman-Hamilton-Jacobi partial differential equation can now be solved.

The familiar three theorems of Rényi concerning the record times in an i.i.d. sequence of random variables are extended to the record times and inter-record times of a sequence of dependent, non-identically distributed random variables defined on a finite Markov chain. These theorems are the Central Limit Theorem (C.L.T.), the Strong Law of Large Numbers (S.L.L.N.) and the Law of the Iterated Logarithm (L.I.L.). Similar results are also obtained for m-record times, inter-m-record times, and for the continuous parameter situation when observations are taken at the epochs of a Poisson process.

The techniques used by Doeblin and Chung to obtain ordinary limit laws (central limit laws, weak and strong laws of large numbers, and laws of the iterated logarithm) for Markov chains, are extended to obtain analogous functional limit laws for stochastic processes which have embedded processes satisfying these laws. More generally, it is shown how functional limit laws of a stochastic process are related to those of a process embedded in it. The results herein unify and extend many existing limit laws for Markov, semi-Markov, queueing, regenerative, semi-stationary, and subordinated processes.

Various attempts have been made to develop a theory of the genetical effects of assortative mating, using various models. The earliest and most famous is that of Fisher (1918). However, this has run into a difficulty in that his paper is far from easy to understand even despite the efforts of Kempthorne (1957) and Moran and Smith (1966) to simplify his presentation. A key paragraph is the following (which we give in abbreviated form).

In the two locus theory that has been developed, it has been assumed that amount of recombination between the two loci is the same in both sexes (Bodmer and Felsenstein, (1967); Karlin and Feldman, (1970)). Some results of assuming that the recombination value in males is rm and in females rf are reported here.

Multidimensional sampling of real data, e.g., in space and time, often requires observations at non-uniformly spaced intervals. A discrete transform of a multidimensional stationary stochastic process transforms a multivariate problem into an asymptotically univariate one if the spacing is uniform in at least one dimension. For both uniform and non-uniform sampling and a model of ‘signal’ imbedded in a ‘noise’ process, asymptotic normality and independence justifies statistical testing in each cell of the transformed domain of the hypothesis ‘noise alone’ versus the alternate ‘signal plus noise’.

In this paper, the authors consider a class of stochastic systems described by Ito differential equations for which both controls and parameters are to be chosen optimally with respect to a certain performance index over a fixed time interval. The controls to be optimized depend only on partially observed current states as in a work of Fleming. However, he considered, instead, a problem of optimal control of systems governed by stochastic Ito differential equations with Markov terminal time. The fixed time problems usually give rise to the Cauchy problems (unbounded domain) whereas the Markov time problems give rise to the first boundary value problems (bounded domain). This fact makes the former problems relatively more involved than the latter. For the latter problems, Fleming has reported a necessary condition for optimality and an existence theorem of optimal controls. In this paper, a necessary condition for optimality for both controls and parameters combined together is presented for the former problems.

For the n-dimensional birth, death and migration process with constant rates, Aksland (1975) found a necessary and sufficient condition that the extinction-probability be one. This condition is translated into the form of direct relations between the parameters of the model.

The three-locus symmetric viability model has been analysed. The symmetric equilibria, namely those with equality of complementary chromosomes and hence with all allele frequencies equal to ½, have been found and their local stability studied. It was shown that there are fifteen symmetric equilibria, of which six may be stable for a given set of selection and recombination parameters.

In the quantitative inheritance of finger-ridge counts, it is conventional to take the largest of the ulnar and radial ridgecounts as a measure of the pattern size. In the case of a loop, there is but one measurement, a radial count on an ulnar loop, and an ulnar count on a radial loop. In a whorl there are (usually) two counts, and it is in this instance when the convention is used. In the case of an arch, both ulnar and radial counts are zero.

The evaluation of the chance of survival of an advantageous mutant gene in an age-structured population is of interest since, as pointed out by Charlesworth (1973), the usual evolutionary arguments concerning the long-term relative effectiveness of natural selection on genes acting at different ages (Cole (1954), Hamilton (1966)) should be translated into statements about the survival probabilities of genes with age-specific effects. We shall be concerned with this quantity for the case of a mutant introduced into a large population with discrete age-classes (Leslie (1945)). Let unity be the index of the birth age-class, b that of the age of first reproduction, and d that of the age of last reproduction. Let px be the probability of survival of individuals from age x to x + 1, and let lx be the probability of surviving from conception to age x (lx = πy=1x-1px). Let mx be half the expected number of offspring at age x (demographic differences between the sexes are assumed to be absent; the environment is assumed constant). Since a new mutant gene is carried largely in heterozygotes for a long time after its first appearance, the problem reduces to that of finding the chance of survival of the line descended from a single individual of age-class 1, with the parameters of the heterozygote for the mutant gene. Let the probability generating function for the distribution of heterozygous offspring by a heterozygote aged x be hx(z). If offspring distributions at different ages are independent, the p.g.f. for the total offspring produced at ages b to x by an individual aged x (x ≧ b) is given as

In the location of genes affecting quantitative characters in Drosophila, Thoday has made extensive use of a technique of examining the products of crossovers between tester chromosomes and those from selected lines. The paper considers the validity of inferences which can be drawn from such experiments. The multiplicity of possible models makes this difficult and some efforts have been made, simulating results from given known situations. It seems probable that the shape of the final conclusions would be in terms of the power of the method to discover groups of genes, covering a certain length of chromosome and with a given sum total of effects.

The relationships between individuals may be specified by the genes which they have in common, where two genes are considered to be the same only if they are identical by descent from some common ancestor. Relationships between two individuals have been extensively studied by, amongst many others, Cotterman (1940), Malécot (1948) and Li and Sacks (1954). Less progress has been made with the more general problem of relationships between an arbitrary number of individuals. Elandt-Johnson (1971) has considered the special case of joint genotype distributions in a sibship, and Hilden (1970) has constructed an algebraic method of combining information on several individuals to give the conditional distribution of a single unborn relative, but neither of these approaches provides a general solution to the problem.

The classical models for genetic drift of Wright (1931) and Moran (1958) have been generalized by Karlin and McGregor (1965), and Chia and Watterson (1969), this last treatment including the other three. The basic technique for the generalized models was to consider the change of certain expectations of the underlying random variables, the generalization being in terms of the conditional branching process.