In a previous investigation (Torrez (1978)) conditions were given for extinction and instability of a stochastic process (Zn) evolving in a random environment controlled by an irreducible Markov chain (Yn) with state space 𝒴 The process (Yn, Zn) is Markovian with state space 𝒴 × {0,1, ···, N} where 𝒴 = {1,· ··,m} and the marginal process (Zn) is a birth and death chain on {0,1,· ··,N}, with 0 and N made absorbing, when conditioned on a fixed sequence of environmental states of (Yn). This paper provides bivariate finite difference methods for calculating (i) P(Zn → 0) when this probability is not one; and (ii) the expected duration of the process Zn. For (i), the cases when the transition probabilities of the (Yn)-conditioned process (Zn) are non-homogeneous and homogeneous are considered separately. Examples are given to illustrate these methods.

This paper examines a model for ecological and epidemiological spread. Expressions are derived for mean waveforms and expectation velocities for two specific contact distributions. Whilst one distribution may be bounded above by a negative exponential function the other may not, and these two situations respectively give rise to finite and infinite asymptotic expectation velocities.

This paper is concerned with the area under the trajectory of carriers in the Downton carrier-borne epidemic (1968). Its Laplace–Stieltjes transform is studied, and a recursive equation derived for it. Its mean is examined in some detail and the solution obtained is expressed as a power series in π, the proportion of the infected susceptibles becoming carriers.

The limit behavior of the stationary distribution of the infinite-alleles model is reduced to a single Laplace transform formula. Some known results, such as Ewens' sampling formula, the distribution of the order-statistics and the frequency spectrum density are shown to follow from this relation. All the results are obtained within the framework of the configuration process, without recourse to finite alleles models.

In view of some recent results by Kingman (1977), the results apply wherever the Ewens' sampling formula is valid.

The change with respect to time of the information in a population is studied, when the population is classified according to genetic type. The cases considered are (i) an infinite population classified at two loci, (ii) a large finite population classified at one locus, subject to sampling fluctuations and (iii) the same, subject to the effect of mutation.

This paper gives the probability that no common allozyme signature is found among n randomly selected cell culture lines when the s mutually exclusive and exhaustive signatures have arbitrary known probabilities. This result is useful in detecting cell culture contamination by a single cell line. This problem is a generalization of the classic problem of finding the probability of no common birthday among n individuals when it is assumed that each individual has a chance 1/s = 1/365 of a birthday on a given day. Both an exact recursive solution and an approximate solution are given for the general problem.

In previous papers (1976), (1977) limit theorems were obtained for the classical Ising model in the absence of an external magnetic field, thereby providing a basis for asymptotic inference. The present paper extends these results to arbitrary external magnetic fields. Statistical inference for this model is important because its nearest-neighbour interactions provide a natural first approximation to spatial interaction among binary variables located on square lattices.

The most interesting behaviour occurs in zero field and at or beyond the critical point. In this case, the central limit result for nearest-neighbour interactions requires an unusual norming, the limiting variances may depend on the nature of the boundary conditions, and there cannot be any central limit result for external magnetic field. The implications of these phenomena for statistical inference are also discussed. In particular, the maximum likelihood estimator of magnetic field is not consistent. Rather it appears to have a non-trivial asymptotic distribution.

A linear DNA molecule may be labelled at a fixed locus by a minute complementary radioactive molecule. A collection of identical molecules is to be so labelled and each one independently cut at a random number, N, of random places, X (N Poisson, X uniform). Fragments containing the label are to be collected and assayed by length. It is shown that the recovery pattern (fragment length distribution) contains a jump discontinuity at the fixed locus and may be used to determine the distance between the attachment site and the nearest end of the molecule.

The recovery pattern under the hypothesis that the collection of molecules are circularly permuted, i.e. the labelled locus is uniformly distributed over the length of the molecule, does not contain such discontinuities. The case where the labelling molecule has a non-negligible extent is also treated.

The fluctuation of the counts of particles in m regions are used to yield an estimate of a fundamental motion parameter. The effect of varying m on the precision of the estimation method is considered. An exact analysis and simulation of the Fürth pedestrian vector process is presented.

Characteristics of some collapsing supercritical stochastic population processes where an embedded Galton–Watson process can be isolated are shown to be identical to those of some associated subcritical process.

It is shown that a phase transition for the Ising model on a tree graph can be detected by the limiting behavior of the magnetization. The relevant limit theorems are special cases of limit theorems for multitype branching processes.

Assume a population is distributed in an infinite lattice of colonies in a migration and random mating model in which all individuals are selectively equivalent. In one and two dimensions, the population tends in time to consolidate into larger and larger blocks, each composed of the descendants of a single initial individual. Let N(t) be the (random) size of a block intersecting a fixed colony at time t. Then E[N(t)] grows like √t in one dimension, t/log t in two, and t in three or more dimensions. On the other hand, each block by itself eventually becomes extinct. In two or more dimensions, we prove that N(t)/E[N(t)] has a limiting gamma distribution, and thus the mortality of blocks does not make the limiting distribution of N(t) singular. Results are proven for discrete time and sketched for continuous time.

If a mutation rate u > 0 is imposed, the ‘block structure' has an equilibrium distribution. If N(u) is the size of a block intersecting a fixed colony at equilibrium, then as u → 0 N(u)/E[N(u)] has a limiting exponential distribution in two or more dimensions. In biological systems u ≈ 10–6 is usually quite small.

The proofs are by using multiple kinship coefficients for a stepping stone population.

We investigate a random collision model for competition between types of individuals in a population. There are dominance relations defined for each pair of types such that if two individuals of different types collide then after the collision both are of the dominant type. These dominance relations are represented by an oriented graph, called a tournament. It is shown that tournaments having a particular form are relatively stable, while other tournaments are relatively unstable. A measure of the stability of the stable tournaments is given in the main theorem.

Cohen has generalized the classical strong ergodic theorem of demography to a stochastic setting. In this setting population projection matrices are chosen according to some homogeneous Markov chain. If this Markov chain converges to the same long-run distribution regardless of its starting point, then one can define an induced Markov chain on the product space of projection matrices and age structure vectors that also has a long-run distribution independent of its starting point. The present paper gives more natural conditions under which Cohen's result holds.

There are some martingales associated with the branching random walk which are natural generalizations of the classical martingale occurring in the Galton–Watson process. Some continuity properties of the distributions of their limits are discussed.

The probability generating functional (p.g.fl.) of a non-homogeneous Poisson cluster process is characterized in Ammann and Thall (1977) via a decomposition of the KLM measure of the process. This p.g.fl. representation is utilized in the present article to show that the family 𝒟 of Poisson cluster processes with a.s. finite clusters is invariant under a class of cluster transformations. Explicit expressions for the finite-dimensional count distributions, product moment measures, and the distribution of clusters are derived in terms of the KLM measure. It is also shown that an element of 𝒟 has no multiple events iff the points of each cluster are a.s. distinct.

In the Poisson disorder problem the probability that the parameter of a Poisson process y, has increased by a constant amount, given observations of ys, s ≦ t, is a Markov process of mixed type with jumps of variable (state-dependent) magnitude superimposed upon a drift which satisfies an ordinary differential equation. Using a likelihood-ratio transformation, one can reduce the backward equation satisfied by the expected first-passage time to a constant level for the mixed process to a differential-difference equation with a constant retardation. We discuss a method for solving this equation and present some numerical results on its solution. The accuracy of some approximations which are easier to calculate is investigated.

The purpose of this note is to show how well the coupling device is fitted for use in the study of birth and death process asymptotics: known achievements get new proofs, and new results on rate of ‘forgetfulness of initial state' and stochastic domination are established.

Most research in first-passage percolation has been done under the assumption of a finite mean for the underlying time coordinate distribution. We demonstrate that the basic ergodic results can be derived under a weaker moment assumption, which still permits us to evaluate the time constant in the case where the atom at zero of the time coordinate distribution exceeds one-half. Further almost sure convergence is investigated more closely.

For some three-dimensional random walks on the cubic lattice, the probability of the walk returning to its starting point is given numerically.