The familiar three theorems of Rényi concerning the record times in an i.i.d. sequence of random variables are extended to the record times and inter-record times of a sequence of dependent, non-identically distributed random variables defined on a finite Markov chain. These theorems are the Central Limit Theorem (C.L.T.), the Strong Law of Large Numbers (S.L.L.N.) and the Law of the Iterated Logarithm (L.I.L.). Similar results are also obtained for m-record times, inter-m-record times, and for the continuous parameter situation when observations are taken at the epochs of a Poisson process.

For the Ek/G/1 queue with finite waiting room the phase technique is used to analyse the Markov chain imbedded in the queueing process at successive instants at which customers complete service, and the distribution of queue length is obtained. The limit as the size of the waiting room becomes infinite is found.

We examine birth and death processes where the birth/death rates depend on the particular configuration of the population. The easiest case is when the configurations form the subsets of some finite lattice (thus the state space is finite). In this case we look at the relationships between time-reversibility, nearest neighbour interactions, and the equilibrium state being a Markov random field. A more interesting case is when the entities which can be born or die can do so at any point in some bounded region of space (or the plane). This gives us a pure jump process, whose general properties are well-known, (see for example, Feller (1966) Vol. II Chapter X.3). We examine some particular examples and compute equilibrium distributions.

This paper (Cohen (1975)) establishes a new class of ergodicity theorems for the age structure of populations, and proposes applications.

We construct an immigration-branching process from an inhomogeneous Poisson process, a parameter-dependent probability distribution of populations and a Markov branching process with homogeneous transition function. The set of types is arbitrary, and the parameter is allowed to be discrete or continuous. Assuming a supercritical branching part with primitive first moments and finite second moments, we prove propositions on the mean square convergence and the almost sure convergence of normalized averaging processes associated with the immigration-branching process.

A continuous-time Markov branching process is modified to allow some dependence of offspring generating function on population size. The model involves a given population size M, below which the offspring generating function is supercritical and above which it is subcritical. Immigration is allowed when the population size is 0. The process has a stationary measure, and an expression for its generating function is found. A limit theorem for the stationary measure as M tends to ∞ is then obtained.

A simple path transformation is described which connects one-dimensional Brownian motion with the radial part of three-dimensional Brownian motion. This provides simple proofs of various path decompositions for these processes described by David Williams.

A shot process with bursts of events is constructed using an occurrence time structure similar to that of clustering point processes. The characteristic function and a simple functional form for the power spectral density of the process are found. It is shown that the burst property might not be observable. Applications of a specific example of this process are given in modeling a 1/f noise commonly found in some electronic components.

Estimation of the properties of particles embedded in a solid medium via observations on random plane sections is studied. The particular properties considered are those associated with steel inclusions, for example, projected area fraction, total caliper length per unit volume and elongation factor. The approach is general, and with the minimum of assumptions about the particle shape, formulae are obtained for suitable estimators and their variances. Generalised moment relations are derived for the case of given, arbitrary shapes. Estimations of the density of particle centres based upon reciprocal observations are shown to lack robustness to small changes in shape and a more reliable density estimator is discussed. Consideration is given to the case of particles with elliptical profiles and to the use of aggregated results which arise when using automatic scanning equipment. Results are given for the case when there is length-biassing of the intercept distribution due to edge-effects for finite sample areas. Finally, the implication of a positive lower resolution point is discussed.

For every N ≧ 1, let be a real-valued stationary process, and let . Suppose that and var, where ∊N and τN are positive null sequences. Limiting distributions of as N → ∞ are obtained for the cases τN = ∊N and τN = o(∊N). These results are established by an extension of a method due to Moran. The theory is illustrated by a variety of applications to genetic models.