Two stochastic models are developed for the predator-prey process. In each case it is shown that ultimate extinction of one of the two populations is certain to occur in finite time. For each model an exact expression is derived for the probability that the predators eventually become extinct when the prey birth rate is 0. These probabilities are used to derive power series approximations to extinction probabilities when the prey birth rate is not 0. On comparison with values obtained by numerical analysis the approximations are shown to be very satisfactory when initial population sizes and prey birth rate are all small. An approximation to the mean number of changes before extinction occurs is also obtained for one of the models.

Iterative sampling procedures of a general type in a finite population are considered. They generalize the Reed-Frost process in that binomial sampling is replaced by an arbitrary symmetric sampling defined by a factorial series distribution. Threshold limit theorems are proved saying that the total number of sampled objects is either small with a certain limit distribution, or a finite fraction of the population with a Gaussian limit distribution as the size of the population gets large. These results extend earlier ones for the Reed-Frost process [1], and are proved in a more direct way than before.

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfy λ0 = 0, λ j > 0 for each j > 0, and . Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λ j = jλ, µj = jμ) with catastrophes of several different types.

The asymptotic final size distribution of a multitype Reed–Frost process, a chain-binomial model for the spread of an infectious disease in a finite, closed multitype population, is derived, as the total population size grows large. When all subgroups are of comparable size, the infection pattern irreducible and the epidemic started by a small number of initial infectives, the classical threshold behaviour is obtained, depending on the basic reproduction rate of the disease in the population, and the asymptotic distributions for small and large outbreaks can be found. The same techniques can then be used to study other asymptotic situations, e.g. small groups in an otherwise large population, large numbers of initial infectives and reducible infection patterns.

A multitype polymer reaction process is considered. Its dynamics is described by means of the evolution in time of the average number of reactions (of any kind) and the mean number of unreacted chain segments (of any length), assuming that the initial chain molecule consists of n unreacted units. The asymptotic behaviour (n →∞) of the variance of the extent of reaction is also studied.

A general exchangeable model is introduced to study gene survival in populations whose size changes without density dependence. Necessary and sufficient conditions for the occurrence of fixation (that is the proportion of one of the types tending to 1 with probability 1) are obtained. These are then applied to the Wright–Fisher model, the Moran model, and conditioned branching-process models. For the Wright–Fisher model it is shown that certain fixation is equivalent to certain extinction of one of the types, but that this is not the case for the Moran model.

A simple proof is given of the result that the ‘overflow' from a finite-state birth and death process is a renewal stream characterized by hyperexponential inter-event times. Our structure is utilized to give a converse result that any hyperexponential renewal stream can be so produced as the overflow from a finite-state birth and death process.

The motion of electrons through a gas of particles is modelled as a two-state process: an electron is alternatingly moving and being held captive by a non-moving particle, during exponentially distributed periods.

The model contains a parameter p, the probability that collision of an electron with a gas particle does not lead to its capture but produces an extra electron.

The quantity of interest is the current C(t) produced by the moving electrons, as a function of time.

Some classes of population-size-dependent Galton-Watson processes are considered where extinction occurs with probability 1. Results on the asymptotic behaviour of the probability of survival up to time t, mean population size and conditioned limit distributions are found to hold. They correspond to those obtained in the study of Galton-Watson processes with immigration stopped at 0.

Let {Xn} be a sequence of independent (or Markov dependent) trials taking values in a given set S. Let JR be a given path of length k in S, i.e. R is a run of length k whose elements come from S. {Xn} may indicate the motion of a particle on S. We consider the problem of finding the probability that at trial m, the particle has for the first time moved length l ≦ k on R which is equivalent to finding the probability of the first occurrence of any subrun of length l ≦ k of R. In the case of l = k this gives the result of Schwager [6].

For a bisexual Galton–Watson branching process with superadditive mating function there is a simple criterion for determining whether or not the process becomes extinct with probability 1, namely, that the asymptotic growth rate r should not exceed 1. When extinction is not certain (equivalently, r > 1), simple upper and lower bounds are established for the extinction probabilities. An example suggests that in the critical case that r = 1, some condition like superadditivity is essential for ultimate extinction to be certain. Some illustrative numerical comparisons of particular mating functions are made using a Poisson offspring distribution.

A stationary stochastic process must satisfy various requirements to make it a realistic model for a phenomenon in the real world. Some of these requirements are quantitative, such as agreement of distribution or moments. Other, more qualitative requirements deal with the general behavior of the process. Two such requirements are non-singularity and asymptotic independence. Each will be discussed from a variety of points of view, and given precise definition in a succession of progressively stronger forms.

Let {Yn: n ≧ 1} be a Harris-recurrent Markov chain on a general state space. It is shown that {Yn} is strong mixing, provided there exists a stationary probability distribution π (·) for {Yn}. Necessary and sufficient conditions for an autoregressive process to be uniform mixing are given.

Linear dynamical systems where both inputs and outputs are contaminated by errors are considered. A characterization of the sets of all observationally equivalent transfer functions is given, the role of the causality assumption is investigated and conditions for identifiability in the case of Gaussian as well as non-Gaussian observations are derived.

McKean's tagged-molecule process is a non-linear homogeneous two-state Markov chain in continuous time, constructed with the aid of a binary branching process. For each of a large class of branching processes we construct a similar process. The construction is carefully done and the weak homogeneity is deduced. A simple probability argument permits us to show convergence to the equidistribution (½, ½) and to note that this limit is a strong equilibrium. A non-homogeneous Markov chain result is also used to establish the geometric rate of convergence. A proof of a Boltzmann H-theorem is also established.

A familiar relation links the densities that result for the intersection of a convex body and straight lines under uniform isotropic randomness with those that result under weighted randomness. An extension of this relation to the intersection of more general domains is utilized to obtain the variance of the n-dimensional measure of the intersection of two bodies under uniform isotropic randomness. The formula for the variance contains the point-pair-distance distributions for the two domains — or the closely related geometric reduction factors. The result is applied to derive the variance of the intersection of a Boolean scheme, i.e. a stationary, isotropic Poisson process of domains, with a fixed sampling region.