The estimation, and testing for the presence, of a selective advantage of one allele over another is considered. It is assumed that a population's allele frequencies are known from some initial time until fixation of one or other allele occurs. The statistics needed to perform the estimation and testing are the heterozygosity of the population summed over all generations, and the observation of which allele fixes. It is shown that certain asymptotic probability distributions arise which are similar to those proved by Brown and Hewitt for statistical inference from diffusion processes, but their results assumed that the diffusion had a stationary density which is not the case for alleles which fix.

The genetic diffusion may be transformed to Brownian motion with constant drift, and the inference questions concerning selection can be transformed to questions about the first exit of a Brownian motion from an interval. It is thus possible to construct significance tests, and to calculate the power of those tests, for detecting selection.

This paper presents limit theorems for the population sizes of a Bienaymé–Galton–Watson process allowing immigration. For the non-critical cases it is known that the limit distribution is non-defective iff a logarithmic moment of the immigration distribution is finite. The new results of this paper are concerned with the situation where this moment is infinite and give limit theorems for a certain slowly varying function of the population size. A parallel discussion is given for the critical case and also for the continuous-time process.

The methods of the paper are used to give some results on the rate of decay of the transition probabilities and on the growth rate of the stationary measure. These in turn are used to obtain some limit theorems for a reversed-time process.

This paper presents some limit theorems for the simple branching process allowing immigration, {Xn}, when the offspring mean is infinite. It is shown that there exists a function U such that {e–nU/(Xn)} converges almost surely, and if s = ∑ bj, log+U(j) < ∞, where {bj} is the immigration distribution, the limit is non-defective and non-degenerate but is infinite if s = ∞.

When s = ∞, limit theorems are found for {U(Xn)} which involve a slowly varying non-linear norming.

We consider the following modification of an ordinary Galton–Watson branching process. If Zn = i, a positive integer, then each parent reproduces independently of one another according to the ith {P(i)k} of a countable collection of probability measures. If Zn = 0, then Zn + 1 is selected from a fixed immigration distribution. We present sufficient conditions on the means μi, the variances σ2i, and the (2 + γ)th central absolute moments β2+γ,i of the {P(i)k}'s which ensure transience of recurrence of {Zn}.

A level crossing predictor is a predictor process Y(t) which can be used to predict whether a specific process X(t) will cross a predetermined level or not. A natural criterion of how good a predictor is can be the probability that a crossing is detected a sufficient time ahead, and the number of times the predictor makes a false alarm.

Suppose the process X(t) consists of a deterministic part A(t) which can be calculated with sufficient accuracy, and a stochastic part Xe(t) which can be predicted by some statistically based predictor An example of this is the prediction of water level near a coast, when A(t) is a sum of known tide components.

The paper develops a tool to handle the detection properties of such predictor processes when used to predict level crossings for the case when A(t) is periodic or is a sum of such functions.

The construction and properties of reversible and dynamically reversible models for finite-state Markov processes are studied. Certain results on approximating processes with rational power spectra with dynamically reversible finite-state models are also obtained.

This paper employs the methods currently used to solve many queuing control models in order to investigate the behavior of a firm's optimal advertising and pricing strategies over time. Given that a firm's market position expands or deteriorates in a probabilistic way which depends upon the current position, the rate of advertising, and the price the firm charges, we present conditions which ensure that the optimal level of advertising is a monotonic function of the firm's market position, and we discuss the economic meaning of these conditions. Furthermore, although the primary focus is upon a non-competitive environment, we develop the above model as a non-zero sum, two-person stochastic game and show that an equilibrium strategy exists which is simple to compute.

A model is developed for the failure time of a bundle of fibers subjected to a constant load. At any time, all surviving fibers share the bundle load equally while all failed fibers support no load. The bundle may collapse immediately or fibers may fail randomly in time, possibly more than one at a time. The failure time of the bundle is the failure time of the last surviving fiber. For a single fiber, the c.d.f. for the failure time is assumed to be a specific functional of an arbitrary load history. The model is developed using a quantile process approach. In the most important case the failure time of the bundle is shown to be asymptotically normal with known parameters. The bundle failure model has the features of both static strength and fatigue failure of earlier analyses, and thus is more realistic than earlier models.

This paper considers undiscounted Markov decision problems. With no restriction (on either the periodicity or chain structure of the problem) we show that the value iteration method for finding maximal gain policies exhibits a geometric rate of convergence, whenever convergence occurs. In addition, we study the behaviour of the value-iteration operator; we give bounds for the number of steps needed for contraction, describe the ultimate behaviour of the convergence factor and give conditions for the existence of a uniform convergence rate.

This paper considers a regular Markov process with continuous parameter, countable state space, and stationary transition probabilities, and defines a class of traffic processes over it. The possibility that multiple traffic processes constitute mutually independent Poisson processes is investigated.

A variety of independence conditions on a traffic process and the underlying Markov process are shown to lead to Poisson-related properties; these conditions include weak pointwise independence, and pointwise independence. Some examples of queueing-theoretic applications are given.

For the class of traffic processes considered here in a queueing-theoretic context, Muntz's MM property, Gelenbe and Muntz's notion of completeness, and Kelly's notion of quasi-reversibility are shown to be essentially equivalent to pointwise independence of traffic and state. The relevance of the theory to queueing network decomposition is pointed out.