The Reed–Frost model for the spread of an infection is considered and limit theorems for the total size, T, of the epidemic are proved in the limit when n, the initial number of healthy persons, is large and the probability of an encounter between a healthy and an infected person per time unit, p, is λ/n. It is shown that there is a critical threshold λ = 1 in the following sense, when the initial number of infected persons, m, is finite: If λ ≦ 1, T remains finite and has a limit distribution which can be described. If λ > 1 this is still true with a probability σm < 1, and with probability 1 – σmT is close to n(1 – σ) and has an approximately Gaussian distribution around this value. When m → ∞ also, only the Gaussian part of the limit distribution is obtained. A randomized version of the Reed–Frost model is also considered, and this allows the same result to be proved for the Kermack–McKendrick model. It is also shown that the limit theorem can be used to study the number of connected components in a random graph, which can be considered as a crude description of a polymerization process. In this case polymerization takes place when λ > 1 and not when λ ≦ 1.

The extinction time distributions of a class of branching processes in varying environments are considered. We obtain (i) sufficient conditions for the extinction probability q = 1 or q < 1; (ii) asymptotic formulae for the tail probability of the extinction time if q = 1; and (iii) upper bounds for 1 – q if q < 1. To derive these results, we give upper and lower bounds for the tail probability of the extinction time. For the proofs, we use a method that compares probability generating functions with fractional linear generating functions.

Conditions are given for which the Markov jump process describing the stochastic model of chemical reactions with diffusion converges to the solution of the corresponding deterministic reaction–diffusion equation.

Let n points in the plane be generated by some specified random mechanism and suppose that N(∊) of the resulting triads form triangles with largest angle ≧ π – ∊. The main object of the paper is to obtain asymptotic formulae for and Var (N(∊)) when ∊ ↓ 0, and to solve the associated data-analytic problem of testing whether an empirical set of n points should be considered to contain too many such ∊-blunt triads in the situation where the generating mechanism is unknown and where all that can be said about the tolerance ∊ is that it must be allowed to take values anywhere in a given interval (T0, T1) (0 < T0 < T1). This problem is solved by the introduction of a plot to be called the pontogram and by the introduction of simulation-based significance tests constructed by random lateral perturbations of the data.

Suppose a smooth surface S is exposed to a penetrating beam of parallel line rays and that, wherever a ray has a tangency with S, there is a corresponding point registration in an image plane H placed perpendicular to the beam. The registering rays form a cylindrical surface which registers in H as a smooth curve C (except possibly for a number of cusps). Properties of C when the beam is taken isotropic random in space are investigated, and stereological applications are noted. The phenomenon of screening of projecting rays, which occurs for example with opaque surfaces exposed to light beams, is considered. Limiting processes permit extensions of the formulae to the cases in which S contains curved edges and in which any boundary curves of S also register. Finally, various related types of random tangent to a surface are considered.

A large class of statistics of planar and spatial data is closely connected with empirical distributions, which estimate ‘ergodic’ distributions of stationary random sets. The main result is a functional limit theorem concerning the deviation of the empirical distribution from the ‘true’ one. Examples in mathematical morphology are given.

The paper derives the asymptotic properties of a class of estimators of the Walsh–Fourier spectral density of a stationary time series. The spectral density is defined in Kohn (1980).

In the present paper a uniform asymptotic series is derived for the probability distribution of the sum of a large number of independent random variables. In contrast to the usual Edgeworth-type series, the uniform series gives good accuracy throughout its entire domain. Our derivation uses the fact that the major components of the distribution are determined by a saddle point and a singularity at the origin. The analogous series for the probability density, due to Daniels, depends only on the saddle point. Two illustrative examples are presented that show excellent agreement with the exact distributions.

Let Xnn denote the largest of n independent N(0, 1) variables. Several methods of estimating P(Xnn ≦ x) are considered. It is shown that X2nn, when normalized in an optimal way, converges to the extreme value distribution at a rate of only 1/(log n)2, and that if 0 < t ≠ 2 then |Xnn|t converges at a rate of 1/log n. Therefore it is not feasible to use the extreme value distribution to estimate probabilities for normal extremes unless the sample size is extremely large. An alternative approach is presented, which gives very good estimates of P(Xnn ≦ x) for n ≧ 10. The case of rth extremes is also considered.

This paper studies the GI/G/1 queueing system in which no customer can stay longer than a fixed interval D. This is also a model for the dam with finite capacity, instantaneous water supply and constant release rule. Using analytical method together with the property that the queueing process ‘starts anew’ probabilistically whenever an arriving customer initiates a busy period, we obtain various transient and stationary results for the system.

Models are given for sequences of correlated exponential interarrival and service times for a single-server queue. These multivariate exponential models are formed as probabilistic linear combinations of sequences of independent exponential random variables and are easy to generate on a computer. Limiting results for customer waiting time under heavy traffic conditions are obtained for these queues. Heavy traffic results are useful for analyzing the effect of correlated interarrival and service times in queues on such quantities as queue length and customer waiting time. They can also be used to check simulation results.

The G/G/1 queueing system is studied by means of the regeneration point method, exploiting the concept of busy cycles. Recurrence relations are set-up for the distribution of the queue length at the arrival epochs. The same method is used to obtain the correlation structure of arrivals and departures for the G/M/1 queue.