The principal aim of this paper is to exhibit applications of techniques of time series analysis for establishing limit distribution theorems of statistical relevance on a subcritical Galton–Watson process X with immigration. In this approach the results obtained by Heyde and Seneta, Quine, and Klimko and Nelson are re-established in a more concise form on adopting new methods of proof, which seek to unify these results. In addition, Quenouille-type limit theorems on X are proved leading to the construction of Quenouille-type goodness-of-fit tests for X. It appears that Billingsley's central limit theorem for martingales is appropriate for proving the basic result, Theorem 1.1. This is done on converting the entire problem as a martingale problem through a use of Lemma 2 of Venkataraman (1968).

The aim of this paper is to investigate the distribution of the extinction times, T, of the stochastic logistic process from both the numerical and the theoretical standpoint. The problem is approached first by deriving formulae for the moments of T; it is then shown that in most cases T is, very nearly, a gamma variate. Some simulated results are given and these agree well with the theory. Furthermore, a consideration of the process conditioned on non-extinction is shown to be an effective way of obtaining a large t (time) description of the unconditioned process. Finally, a more general form of the model in which the death-rate as well as the birth-rate is ‘density-dependent' is considered, and by comparison with the usual form of the model the effect on T of this additional factor is assessed.

We introduce a class of infinite-dimensional diffusion processes which contains a limiting version of the Ohta–Kimura model in population genetics. For this a necessary and sufficient condition for existence of stationary distributions is obtained. We are especially interested in the case where there is no stationary distribution. Then it is shown that an individual ergodic theorem holds for a suitably centralized process. As a corollary the wandering distribution exists.

Let a Poisson process be observed whose output rate is one of two levels given by the state of an unseen Markov process. If one of the levels is 0, a simple formula is given for the best guess of the state at any instant based on the stream of past Poisson events. In other cases bounds are given for the likelihood ratio of the state probabilities given the event stream.

We consider Markov models for growth of populations subject to catastrophes. Emphasis is placed on discrete-state models where immigration is possible and the catastrophe rate is population-dependent. Explicit formulas for descriptive quantities of interest are derived when catastrophes reduce population size by a random amount which is either geometrically, binomially or uniformly distributed. Comparison is made with continuous-state Markov models in the literature in which population size evolves continuously and deterministically upwards between random jumps downward.

This paper deals with the parity of individuals in an age-dependent birth and death process. A more general model with parity and age-dependent birth rates is also considered. The mean number of individuals with parity 0, 1, 2, ·· ·is obtained for the two models. The first moments of the total number of births in the population up to time t and the sum of the parities of the individuals existing at time t are obtained. A brief discussion on the parity of individuals in a population including ‘twins' is also given.

Bounds of exponential type are derived for the first-hitting time and occupation times of a real-valued random sequence which has a uniform negative drift whenever the sequence is above a fixed level. The only other assumption on the random sequence is that the increments satisfy a uniform exponential decay condition. The bounds provide a flexible technique for proving stability of processes frequently encountered in the control of queues.

Two applications are given. First, exponential-type bounds are derived for a GI/G/1 queue when the service distribution is exponential type. Secondly, geometric ergodicity is established for a certain Markov chain in which arises in the decentralized control of a multi-access, packet-switched broadcast channel.

Let be a supercritical Bellman-Harris process with finite offspring mean. Cohn [17] has shown that there always exist constants Ct such that limt→∞Zt/Ct = W almost surely for some non-degenerate random variable W. In this paper we give an alternative proof, based on the study of (Zt) as a point process. Our methods are to some extent analytical and parallel Seneta's [18] and Heyde's [11] approaches in the case of the Galton–Watson process. We further identify Ct as 1/(–log Ft(–1)(γ)), where Ft(γ) = E(γzt), i.e. the norming constants found by Seneta [18] for the Galton–Watson process, apply also to the Bellman-Harris process. Finally we derive a weak law of large numbers for W, prove that W is continuous on (0,∞) and show that W has [0,∞) as its support.

In order to study the transience of Hendricks libraries, we introduce and study a special class of Markov chains, the Tsetlin d-piles, generalizing Tsetlin libraries and briefly defined as follows: a 1-pile is a Tsetlin library and a d-pile is a Tsetlin library where each book is replaced by a (d − 1)-pile. We give a stationary measure of these chains and establish the necessary and sufficient conditions for positive recurrence and transience. Finally, the study of d-piles allows us to determine a sufficient condition for transience of quite a large class of Hendricks libraries.

Books being placed on a shelf, a book bi is selected with probability pi, it is removed and replaced at the left-hand end of the shelf prior to next removal. The different arrangements of books on the shelf are considered as states in a Markov chain, known as a Tsetlin library. In this paper, we establish the necessary and sufficient condition of transience for some extensions of this chain.

Let

where Xi, 1 ≦ i ≦ n, are i.i.d. and uniformly distributed in [0, 1]2. It is proved that Mn ∽ cn1–p/2 a.s. for 1 ≦ p <2. This result is motivated by recent developments in the theory of algorithms and the theory of subadditive processes as well as by a well-known problem of H. Steinhaus.

In fitting genetic models on the basis of observations on an interrelated structure, sequential procedures can enhance efficiency. In this paper we consider the case of a rare gene segregating in a single large pedigree. The sampling rule is dictated by the effect of observations on the genotypic probability distribution of unobserved relatives; this effect is investigated.

In this paper, we investigate the behaviour of a Gaussian random field after an ‘upcrossing' of a particular level. In Section 1, we briefly discuss model processes and their background, and give a definition of an upcrossing of a level for random fields. A model field is constructed for the random field after an upcrossing of a level by using horizontal window conditioning in Section 2. The final section contains asymptotic distributions for the model field and for the location and height of the ‘closest' maximum to the upcrossing as the level becomes arbitrarily high.

The focus of this paper is on obtaining a conservative but tight bound on the probability distribution for the strength of a fibrous material. The model is the chain-of-bundles probability model, and local load sharing is assumed for the fiber elements in each bundle. The bound is based upon the occurrence of two or more adjacent broken fiber elements in a bundle. This event is necessary but not sufficient for failure of the material. The bound is far superior to a simple weakest link bound based upon the failure of the weakest fiber element. For large materials, the upper bound is a Weibull distribution, which is consistent with experimental observations. The upper bound is always conservative, but its tightness depends upon the variability in fiber element strength and the volume of the material. In cases where the volume of material and the variability in fiber strength are both small, the bound is believed to be virtually the same as the true distribution function for material strength. Regarding edge effects on composite strength, only when the number of fibers is very small is a correction necessary to reflect the load-sharing irregularities at the edges of the bundle.

Recursive estimation is considered for parameters of certain continuous stochastic models. Several optimality properties are shown to hold for the resulting recursive estimator, where a stochastic approximation viewpoint is taken when deriving statistical properties, like strong consistency and convergence in distribution. Applications are considered throughout, where for example explosion theory for diffusion processes is used as a modeling guide, in a particular application.