A Markov chain model related to population genetics and its convergence to a diffusion process on the multi-dimensional bounded domain are treated. We discuss the case where natural selection is random and the different selection effects over successive generations are independent. Our model is a multi-allelic version of the haploid model of Karlin and Levikson. The asymptotic properties of the limiting diffusion are stated.

A formula is proved for the expected value of sum of a function of gene frequency along a sample path in the past, given the present frequency. The proof and the explanation of the formula is based on the general theory of time reversal of Markov processes. Moreover, a relation between time reversal and conditional processes is discussed, and it is shown that the fictitious drift term appears when one looks back at the history of mutants given the present frequency.

Correction terms are obtained for the diffusion approximation to one- and two-barrier ruin problems in finite and infinite time. The corrections involve moments of ladder-height distributions, and a method is given for calculating them numerically. Examples show that the corrected approximations can be much more accurate than the originals.

Various classifications and representations of multilocus recombination structures are delimited. A class of recombination distributions called the count–location chiasma process is parametrized by a distribution of the number of crossover events and for such crossover events by a conditional distribution of crossover locations. A number of properties and examples of this recombination structure are developed connecting orderings among the recombination mapping functions and the nature of interference.

In one version of the familiar ‘secretary problem’, n rankable individuals appear sequentially in random order, and a selection procedure (stopping rule) is found to minimize the expected rank of the individual selected. It is assumed here that, instead of being a fixed integer n, the total number of individuals present is a bounded random variable N, of known distribution. The form of the optimal stopping rule is given, and for N belonging to a certain class of distributions, depending on n, and such that E(N) → ∞ as n → ∞, some asymptotic results concerning the minimal expected rank are given.

An axiomatic derivation of the Wright–Fisher–Kimura (wfk) diffusion model for genetic drift is given using a variational principle. This is analogous to the characterization of the standard normal distribution in terms of an isoperimetric problem in the calculus of variations where the integrand is Shannon's information measure. We, on the other hand, give a Fisher-type information-theoretic interpretation of the variational principle largely motivated by the geometric approach to statistical likelihood theory due to S. V. Huzurbazzar, B. R. Rao and A. W. F. Edwards. In our process theory, the Ricci curvature tensor plays the role of information matrix. Ultimately, it is proved to be a normalized matrix of second partial derivates of the pseudodensity associated with the Christoffel velocity field. The proofs use classical projective differential geometry and depend on previous work in this series on the geometry of random drift.

We consider an individual which performs a diffusive motion in a certain state space and dies according to a state-dependent death rate. An integral equation for the survival probability is derived, and finally a sufficient criterion for the existence of an initial state is given, for which the corresponding individual dies with probability one.

Statistical inference for Markov processes is commonly based on the maximum likelihood method of estimation and the likelihood ratio criterion for testing hypotheses. Construction of estimators and test statistics by these methods require that a model be chosen in the form of a family of transition density functions. In this paper, asymptotic properties of the maximum likelihood estimator and of the likelihood ratio statistic λn are examined when the model chosen for their construction is incorrect—that is, when no density in the model is a density for the transition probability distribution of the Markov process. It is shown that if and λn are constructed from a ‘regular’ incorrect model, then is consistent and asymptotically normally distributed and the asymptotic null distribution of −2 log λn is that of a linear combination of independent chi-squared random variables. These results are applied to propose measures of the performance of the test based on λn when the statistic is constructed from an incorrect model.

The present paper considers the stochastic difference equation Yn = AnYn-1 + Bn with i.i.d. random pairs (An, Bn) and obtains conditions under which Yn converges in distribution. This convergence is related to the existence of solutions of and (A, B) independent, and the convergence w.p. 1 of ∑ A1A2 ··· An-1Bn. A second subject is the series ∑ Cnf(Tn) with (Cn) a sequence of i.i.d. random variables, (Tn) the sequence of points of a Poisson process and f a Borel function on (0, ∞). The resulting random variable turns out to be infinitely divisible, and its Lévy–Hinčin representation is obtained. The two subjects coincide in case An and Cn are independent, Bn = AnCn, An = U1/αn with Un a uniform random variable, f(x) = e−x/α.

The effect on cable reliability of random cyclic loading such as that generated by the wave-induced rocking of ocean vessels deploying these cables is examined. A simple model yielding exact formulas is first explored. In this model, the failure time of a single fiber under a constant load is assumed to be exponentially distributed, and the random loadings are a two-state stationary Markov process. The effect of load on failure time is assumed to follow a power law breakdown rule. In this setting, exact results concerning the distribution of bundle or cable failure time, and especially the mean failure time, are obtained. Where the fluctuations in load are frequent relative to bundle life, such as may occur in long-lived cables, it is shown that randomness in load tends to decrease mean bundle life, but it is suggested that the reduction in mean life often can be restored by modestly reducing the base load on the structure or by modestly increasing the number of elements in the bundle.

In later pages this simple model is extended to cover a broader range of materials and random loadings. Asymptotic distributions and mean failure times are given where fibers follow a Weibull distribution of failure time under constant load, and loads that are general non-negative stationary processes subject only to some mild condition of asymptotic independence. When the power law breakdown exponent is large, the mean time to bundle failure depends heavily on the exact form of the marginal probability distribution for the random load process and cannot be summarized by the first two moments of this distribution alone.