In a random sample of n pairs (Xr, Yr), r = 1, 2, …, n, drawn from a bivariate normal distribution, let Xr:n be the rth order statistic among the Xr and let Y[r:n] be the Y-variate paired with Xr:n. The Y[r:n], which we call concomitants of the order statistics, arise most naturally in selection procedures based on the Xr:n. It is shown that asymptotically the k quantities k fixed, are independent, identically distributed variates. In addition, putting Rt,n for the number of integers j for which , the asymptotic distribution and all moments of n–1Rt, n are determined for t such that t/n → λ with 0 < λ < 1.

The product moment equations previously derived are first discussed for the infinite one- and two-dimensional Ising (or autologistic) model in the case of non-zero mean, as a prelude to an examination of the probability structure in the higher-dimensional (and nominally zero mean) case below the ‘critical temperature’. Of two simple possible models, A and B, both consistent with the division of the product moment p into ergodic, and long-range non-ergodic, components, such that ρ = r (1 – m2) + m2, where r is the intrinsic correlation coefficient, it is shown that the second model B appears appropriate to the three-dimensional ‘spherical model’, but the first model A to the Ising model. Model A is defined by xi = yi + M, where M = +m or –m, and E{yi} = 0; and Model B by

Necessary and sufficient conditions are obtained for the weak convergence of the sample range of i.i.d. random variables as the number of observations tends to infinity.

Controls of the perturbation type are evolved for the purpose of monitoring the random movement of a response curve using noise-corrupted observations of the response. Dynamic programming methods are used to obtain approximations to the optimum control system when the response is quadratic. Further results are briefly described.

It is shown that all stationary self-exciting point processes with finite intensity may be represented as Poisson cluster processes which are age-dependent immigration-birth processes, and their existence is established. This result is used to derive some counting and interval properties of these processes using the probability generating functional.

A continued fraction representation is presented of the Laplace transform of the generating function of the fundamental joint probability and density of busy period length measured in customers served and duration in time. The setting is the single server Erlang queueing system where the parameters of negative exponentially distributed arrival and service times have a general dependence on instantaneous system state.

The paper deals with a stochastic organization model described by a generalized Poisson input process and given probabilities of entering individuals obtaining certain attributes in this organization. The distribution of attributes at any given instant is derived. Examples illustrate the application of obtained results.

The recurrent epidemic model has hitherto proved mathematically intractable. It is shown here that by imposing the mild condition that the source of new susceptibles is initially of a constant size M for M sufficiently large that there is zero probability of more than M new susceptibles being added to the population in the course of the epidemic, then state probabilities for the stochastic recurrent epidemic model can be found.

In many endemic areas the level of infection of malaria exhibits a regular cycle. A stochastic model is formulated to describe endemic malaria and is used to justify a set of deterministic equations. The model incorporates the properties peculiar to malaria, i.e., in endemic areas malaria is not usually fatal and an individual can be and usually is reinfected. The theory of small oscillations is applied to the deterministic equations, and under the assumption of stable populations of hosts and vectors an expression is obtained for the period of oscillation of the infection.

The geometric distribution is characterized. The lack of memory property is replaced by the constancy of the conditional variance. Then the characterization is obtained.

A truncated point process is defined in terms of a point process and a truncation mechanism independent of the process. The equivalence between two different types of truncation is shown, where the truncations occur after a geometric number of events have occurred or after an exponential time has elapsed, some examples are given.

In a multitype critical age dependent branching process with immigration, the numbers of cell types born by t, divided by t2, tends in law to a one-dimensional (degenerate) law whose Laplace transform is explicitily given. The method of proof makes a correspondence between the moments in the m-dimensional case and the one-dimensional case, for which the corresponding limit theorem is known. Other applications are given, a possible relaxation of moment assumptions, and extensions are indicated.

The accurate space derivative (ASD) method for the numerical treatment of nonlinear partial differential equations has been applied to the solution of Fisher's equation, a nonlinear diffusion equation describing the rate of advance of a new advantageous gene, and which is also related to certain water waves and plasma shock waves described by the Korteweg-de-Vries-Burgers equation. The numerical experiments performed indicate how from a variety of initial conditions, (including a step function, and a wave with local perturbation) the concentration of advantageous gene evolves into the travelling wave of minimal speed. For an initial superspeed wave this evolution depends on the cutting off of the right-hand tail of the wave, which is physically plausible; this condition is necessary for the convergence of the ASD method. Detailed comparisons with an analytic solution for the travelling waves illustrate the striking accuracy of the ASD method for other than very small values of the concentration.

Let X1,n ≦ … ≦ Xn, n be the order statistics of a sample of size n from a distribution function F. Desu (1971) showed that if for all n ≧ 2, nX1,n is identically distributed as X1, 1, then F is the exponential distribution (or else F degenerates). The purpose of this note is to point out that special cases of known characterization theorems already constitute an improvement over this result. We show that the characterization is preserved if “identically distributed” is weakened to “having identical (finite) expectation”, and “for all n ≧ 2” is weakened to “for a sequence of n's with divergent sum of reciprocals”.

A model is studied in which each of several servers assembles finished products consisting of N different input items. Items of each type arrive independently at the assembly station and are grouped into N-tuples consisting of one item of each type. N-tuples are assembled into finished products by the servers on a first come-first-served basis. The model is analyzed by means of the theory of weak convergence, and functional limit theorems are obtained for appropriately normalized random functions induced by the queue size processes. The limits are expressed as functionals of multi-dimensional Wiener processes, with ordinary central limit theorems obtained as corollaries.

With each individual in a branching population associate a random function of the age. Count the population by the values of these functions. Different choices yield different processes. In the supercritical case a unified treatment of the asymptotics is possible for a wide class, including for example the number of individuals having some random age dependent property or integrals of branching processes. As an application, the demographic concept of average age at childbearing is given a rigorous interpretation.

Some minor corrections to Heyde and Seneta (1972) are made, and new convergence rate results given. Estimation by recurrence methods is discussed, as announced earlier.

A detailed probabilistic treatment is given of a birth-and-death process proposed by Williams (1969) in which the elements of the process bear up to s inanimate marks. Equations for the second-order moments of the process, and approximate marginal univariate solutions, are derived. The exact bivariate solution is given for the case s = 1. For general s the variance of the mark population is also derived.