The branching diffusion with immigration is studied. Under general branching and diffusion laws, the process is shown to be mixing, according to Brillinger's definition. Brillinger's central limit theorem for spatially homogeneous mixing processes is generalized to prove that, under a renormalization transformation, the distribution of the branching diffusion with immigration converges to a completely random Gaussian random measure. In addition, the existence of a steady-state distribution is proven in the case of subcritical branching, and this distribution is shown to be mixing. Hence the steady-state random field also obeys a spatial central limit theorem.

The limiting behavior of the probability of extinction of critical age-dependent branching processes with generation dependence is obtained using Goldstein's methods. Regularity conditions on the mean and variance of the birth distributions are assumed. Also the lifespan distribution is assumed to satisfy suitable regularity conditions.

A simple epidemic process in which the number of individuals who can become infected at any point in time is itself a random variable is described. The discrete asymptotic behaviour of such a process is discussed. In particular, the associated marginal distribution of the limiting process is considered.

The connection between the age distribution of a discrete-time Markov chain and a certain time-reversed Markov chain is exhibited. A method for finding properties of age distributions follows simply from this approach. The results, which have application in several areas in applied probability, are illustrated by examples from population genetics.

We prove in the setting of the system of unbounded spins developed by Lebowitz and Presutti, a strong variational principle analogous to Lanford and Ruelle's for discrete spins. We establish this result for finite-range regular and superstable potentials satisfying a condition which enables us to define the thermodynamic energy of the system.

The asymptotic behavior of the maximum likelihood estimators of Markov models or autoregressive models are given when the true distribution is not a member of the assumed parametric family. The derivation of Akaike's Information Criterion is reviewed for this case.

Let g(x) be the spectral density of a Gaussian stationary process. Then, for each continuous function ψ (x) we shall give an estimator of whose asymptotic variance is O(n–1), where Φ(·) is an appropriate known function. Also we shall investigate the asymptotic properties of its estimator.

Non-linear time series models for non-linear vibrations are presented. Some typical behaviour of non-linear vibrations generated from Duffing's equation or van der Pol's equation are explained through the models.

Given a strictly stationary sequence {Xk, k = …, −1,0,1, …} of r.v.'s one defines for n = 1, 2, 3 …, . Here an example of {Xk} is given with finite second moments, for which Var(X1 + … + Xn)→∞ and ρ n → 0 as n→∞, but (X1 + … + Xn) fails to be asymptotically normal; instead there is partial attraction to non-stable limit laws.

We obtain a representation of an arbitrary discrete distribution with n mass points by an equiprobable mixture of r distributions, each of which has no more than a (≧2) mass points, where r is the smallest integer greater than or equal to (n – 1)/(a – 1). An application to the generation of discrete random variables on a computer is described, which has as an important special case Walker's (1977) alias method.

One probability measure is less than or equal to another in the sense of UCSO (uniform conditional stochastic order) if a standard form of stochastic order holds for each pair of conditional probability measures obtained by conditioning on appropriate subsets. UCSO can be applied to the comparison of lifetime distributions or the comparison of decisions under uncertainty when there may be reductions in the set of possible outcomes. When densities or probability mass functions exist on the real line, then the main version of UCSO is shown to be equivalent to the MLR (monotone likelihood ratio) property. UCSO is shown to be preserved by some standard probability operations and not by others.

Sufficient conditions are derived for ergodicity of the two-dimensional process of binary variables discussed by Galbraith and Walley (1976).

Let L(t) be the random variable which represents the minimum of length of gaps generated by random packing of unit intervals into [0, t]. We have with Using this equation the asymptotic behaviour of P(L(x)≧h) is discussed.

Many random variables arising in problems of geometric probability have intractable densities, and it is very difficult to find probabilities or percentage points based on these densities. A simple approximation, a generalization of the chi-square distribution, is suggested, to approximate such densities; the approximation uses the first three moments. These may be theoretically derived, or may be obtained from Monte Carlo sampling.

The approximation is illustrated on random variables (the area, the perimeter, and the number of sides) associated with random polygons arising from two processes in the plane. Where it can be checked theoretically, the approximation gives good results. It is compared also with Pearson curve fits to the densities.

Starting from the theory of point processes the concept of a process with an embedded marked point process is defined. It is shown that the known formula expressing the relation between the stationary and synchronous version of a regenerative process remains valid without the assumption of independence of cycles. General formulae for stationary availability and interval reliability of complex systems with repair are also obtained. In this way generalizations of Keilson's results for Markovian systems and Ross's results for systems with separately maintained elements are presented. The formulae are applied to a two-unit parallel system with a single repair facility.

In this article we examine a breakdown model in which the system's status can be determined only by a test. Upon detection of failure the system must be replaced by a new identical one. The costs incurred include cost of inspection, operating costs, failure cost and a cost associated with planned replacements. Throughout the paper we restrict attention to replacement rules in which the time interval between two successive inspections is regarded as a fixed quantity. The decision variables include the inspection interval and the scheduling for preventive (planned) replacements. The problem is to specify a replacement rule which minimizes the long-run average cost per unit time. We show that under certain monotone conditions there is a natural candidate for an optimal replacement rule.

In this paper it is shown that for a large class of replacement problems the class of stationary replacement strategies is complete, i.e. in order to minimize the average long run cost per unit time it suffices to consider replacement rules which are equal for each new unit irrespectively of what has been observed from earlier units. The main result is based on a version of the law of large numbers for martingale differences proved in the appendix.

We consider preemptive scheduling of N tasks on m processors; processors have different speeds, tasks require amounts of work which are exponentially distributed, with different parameters. The policies of assigning at every moment the task with shortest (longest) expected processing time among those not yet completed to the fastest processor available, second shortest (longest) to the second fastest etc., are examined, and shown to minimize expected values of various cost functions. As special cases we obtain policies which minimize expected flowtime, expected makespan and expected lifetime of a series system with m component locations and N spares.

In teletraffic measurements, a call arrival process is commonly studied using a method with time-uniform or periodic scanning. The information recorded is the number of calls arrived between the scannings, from which data the number of scans between two successive calls is obtained. These later numbers are used as a measure of the interarrival times.

For an exponential call arrival process, except in the case of Poisson scanning, all other scanning schemes yield the number of scans which are not independent in any two interarrival intervals. By treating the problem as an interaction of two stationary stochastic point processes, we determine the exact joint probability distribution of the number of scans in two adjacent and non-adjacent interarrival intervals. An explicit expression for the correlation coefficient is also obtained.

Level crossings in a stationary dam process with additive input and arbitrary release are considered and an explicit expression for the expected number of downcrossings (and also overcrossings) of a fixed level, per time unit, is obtained. This leads to a short derivation of a basic relation which the stationary distribution of a general dam must satisfy.