A model for the spread of a carrier-borne disease is considered which allows for individual variability. The solutions are used to study the effect of relaxing the assumption of ‘homogeneous mixing’. It is shown that the mean size of an epidemic outbreak is a minimum when the individuals are homogeneous.

A quantum-mechanical central limit theorem for sums of pairwise anti-commuting representations of the canonical anti-commutation relations over a finite-dimensional space is formulated and proved.

Very general forms of the strong law of large numbers and the central limit theorem are proved for estimates of the unknown parameters in a sinusoidal oscillation observed subject to error. In particular when the unknown frequency θ0, is in fact 0 or <it is shown that the estimate, , satisfies for N ≧ N0 (ω) where N0 (ω) is an integer, determined by the realisation, ω, of the process, that is almost surely finite.

For a Markov process in discrete time, having finite covariances, both the transient behaviour and the correlation structure may be summed up in a single generating function, here called the key function. If the process is stationary, and the key function possesses a suitable analytic extension, then the process possesses a continuous spectral density, which can be calculated from the key function. A converse result, and some results for the non-stationary process, are also obtained. The calculation is discussed in more detail for the waiting-time process in the GI/G/1 queue.

{Xn, n ≧ 1} are i.i.d. unbounded random variables with continuous d.f. F(x) =1 —e –R(x). Xj is a record value of this sequence if Xj >max {X1, …, Xj-1} The almost sure behavior of the sequence of record values {XLn} is studied. Sufficient conditions are given for lim supn→∞XLn/R–l(n)=ec, lim inf n → ∞XLn/R−1 (n) = e−c, a.s., 0 ≦ c ≦ ∞, and also for lim supn→∞ (XLn—R–1(n))/an =1, lim infn→∞ (XLn—R–1(n))/an = − 1, a.s., for suitably chosen constants an. The a.s. behavior of {XLn} is compared to that of the sequence {Mn}, where Mn = max {X1, …, Xn}. The method is to translate results for the case where the Xn's are exponential to the general case by means of an extended theory of regular variation.

First, asymptotic results for inter-record times when the CDF of the underlying IID process is not necessarily continuous are obtained, by a stochastic order argument, from known results for the continuous case. Then the asymptotic behaviour of the bivariate process of upper-record values and inter-record times is studied. Finally, assuming continuity of the underlying CDF, we derive the law of the process of total times spent in sets of states, viewing upper record values as states and inter-record times as times spent in a state, the process so viewed being a discrete time continuous state Markov jump process.

The possible relevance of this result to single lane road traffic flow is indicated.

Probability distributions relating to the number and positions of lower records in a finite sequence of observations are obtained by methods involving the permutation of ranks. Some new results, and results of earlier authors, are expressed in a form thought to be useful for direct practical applications.

Expressions are also given for the distribution of the length of a ‘record run’, which consists of the observations following and including a given record either up to the next record, or, if a further record does not occur, up to the end of the sequence of observations. This quantity, whose distribution is closely related to that of the inter-record time, has a direct and simple application to road traffic leaving a signal which has been confirmed by experimental observation.

The usual definition of stochastic comparison of random vectors is extended to stochastic comparison of random processes. Conditions are stated under which {X (t), t ≧ 0} stochastically larger than {Y (t), t ≧ 0} implies that for increasing functionals f.

Applications are made to reliability problems, yielding stochastic comparisons for systems of independently operating machines assuming exponential failure and exponential repair. From these stochastic comparisons we may then deduce similar stochastic comparisons for functionals of practical importance in reliability applications, such as the total machine up-time, the first time that the number of functioning machines drops below a specified number, the total time during which at least a specified number of machines are functioning, etc.

Based on a Wiener process approximation, a sequential test for the bundle strength of filaments is proposed and studied here. Asymptotic expressions for the OC and ASN functions are derived, and it is shown that asymptotically the test is more efficient than the usual fixed sample size procedure based on the asymptotic normality of the standarized form of the bundle strength of filaments, studied earlier by Daniels (1945), and Sen, Bhattacharyya and Suh (1973).

Nonzero-sum N-person stochastic games are a generalization of Shapley's two-person zero-sum terminating stochastic game. Rogers and Sobel showed that an equilibrium point exists when the sets of states, actions, and players are finite. The present paper treats discounted stochastic games when the sets of states and actions are given by metric spaces and the set of players is arbitrary. The existence of an equilibrium point is proven under assumptions of continuity and compactness.

Stationary processes which are defined on the points of a square lattice and are Markovian in various senses are considered. It is shown that a certain assumption of linearity of regression forces the spectral distribution to be of a certain explicit form, and that given this form Gaussian processes of this kind are easily constructed. Certain non-Gaussian processes satisfying the various Markovian properties are also constructed and the difference from nearest-neighbour systems emphasized. It is conjectured, but not proved, that the assumption of linearity of regression also implies Gaussianity.

A tandem queue with K single server stations and unlimited interstage storage is considered. Customers arrive at the first station in a renewal process, and the service times at the various stations are mutually independent i.i.d. sequences. The central result shows that the equilibrium waiting time vector is distributed approximately as a random vector Z under traffic conditions (meaning that the system traffic intensity is near its critical value). The weak limit Z is defined as a certain functional of multi-dimensional Brownian motion. Its distribution depends on the underlying interarrival and service time distributions only through their first two moments. The outstanding unsolved problem is to determine explicitly the distribution of Z for general values of the relevant parameters. A general computational approach is demonstrated and used to solve for one special case.

We study a transportation system consisting of S vehicles of unit capacity and N passenger terminals. Customers arrive stochastically at terminal i, 1 ≦ i ≦ N, seeking transportation to a terminal j, 1 ≦ j ≦ N, with probability Pij. Customers at each terminal are served as vehicles become available. Each vehicle is dispatched from a terminal when loaded, whereupon it travels to the destination of its passenger, according to a stochastic travel time. It is shown under mild conditions that the system is unstable, due to random fluctuations of independent customer arrival processes. We obtain limit theorems, in certain special cases, for the customer queue size processes. Where a steady-state limit exists, this limit is expressed in terms of the corresponding limit in a related GI/G/S queue. In other cases, functional central limit theorems are obtained for appropriately normalized random functions.

The use of a branching process argument in complex queueing situations often leads to a discussion of a non-linear matrix integral equation of Volterra type. By the use of a fixed point theorem we show these equations have a unique solution.

The mean square and almost sure convergence of W(t) = e–αtZ(t) is proved for a super-critical multitype age-dependent branching process allowing immigration at the epochs of a renewal process. It is shown that the Malthusian parameter, asymptotic frequencies of types and stationary age distributions are the same for the processes with and without immigration.

R-positivity theory for Markov chains is used to obtain results for random environment branching processes whose environment random variables are independent and identically distributed and whose environmental extinction probabilities are equal. For certain processes whose eventual extinction is almost sure, it is shown that the distribution of population size conditioned by non-extinction at time n tends to a left eigenvector of the transition matrix. Limiting values of other conditional probabilities are given in terms of this left eigenvector and it is shown that the probability of non-extinction at time n approaches zero geometrically as n approaches ∞. Analogous results are obtained for processes whose extinction is not almost sure.

If the information content of a complete finite probability scheme is greater than that of another such scheme in one measure of information, it seems reasonable to expect that this relation remains true in any other valid measure. In this paper Shannon and Rényi measures are discussed, regarding this aspect.

Studies of storage capacity of reservoirs, under the assumption of infinite storage, lead to the problem of finding the distribution of the range or adjusted range of partial sums of random variables.

In this paper, formulas for the expected values of the range and adjusted range of partial sums of exchangeable random variables are presented. Such formulas are based on an elegant result given in Spitzer (1956). Some consequences of the aforementioned formulas are discussed.

The Poisson process enjoys two special properties: the mean forward recurrence time at time t does not depend on t, and the mean backward recurrence time at time t is the “mean” of the interval distribution truncated at t. Poisson process is the only renewal process with these properties.

A stationary Gaussian process is exhibited with the following property: the covariance function of the process is not differentiable at the origin and yet almost all the sample paths of the process are differentiable in a set of points of the power of the continuum. The process provides a counter example to a statement of Slepian.