A formal proof is given of the equivalence, conjectured by Rostron, of two measures of the genetic difference between two populations.

A vector X of patient prognostic variables is modeled as a linear diffusion process with time-dependent, non-random, continuous coefficients. The instantaneous force of mortality (hazard function) operating on the patient is assumed to be a time-dependent, continuous quadratic functional of the prognostic vector. Conditional on initial data X0, the probability of surviving T units of time is expressed in terms of the solution of a Riccati equation, which can be evaluated in closed form if the coefficients of the process and the hazard are constant. This conditional expectation does not preserve proportional hazards.

For distributions of non-negative random variables with a monotonic hazard rate, it is well known that the normalised moments have the same sign-regular property as the distribution. This note extends this correspondence in properties for some common distributions with non-monotonic hazard rates, linking a change in sign-regular properties.

Using Slepian processes as a standard of comparison, estimates are given for the probability that a centered multiparameter stationary Gaussian process reaches a constant barrier u on a subset of the parameter domain.

Models have been proposed in many diverse areas to generate a lognormal distribution and the underlying idea has always been some form of the law of proportionate effect. In a sense any model must resemble this recipe: take logs and apply the central limit theorem. Our model is no exception. However our formulation is designed to encompass the previous models and demonstrate that the fundamental concept is one of classification. This supersedes a multitude of models which incorporate a mechanism peculiar to a specific application in order to use the law of proportionate effect; we illustrate with applications of the model to ecology, econometrics and geostatistics.

In recent papers, McLeish and others have obtained invariance principles for weak convergence of martingales to Brownian motion. We generalize these results to prove that solutions of discrete-time stochastic difference equations defined in terms of martingale differences converge weakly to continuous-time solutions of Ito stochastic differential equations. Our proof is based on a theorem of Stroock and Varadhan which characterizes the solution of a stochastic differential equation as the unique solution of an associated martingale problem. Applications to mathematical population genetics are discussed.

Suppose that a device is subjected to shocks and that denotes the probability of surviving k shocks. Then is the probability that the device will survive beyond t, where is the counting process which governs the arrival of shocks. A-Hameed and Proschan (1975) considered the survival function H(t) under what they called the pure birth shock model. In this paper we shall prove that is IFRA and DMRL under conditions which differ from those used by A-Hameed and Proschan (1975).

The subject of this paper is networks of queues with an infinite number of servers at each node in the system. Our purpose is to point out that independent motions of customers in the system, which are characteristic of infinite-server networks, lead in a simple way to time-dependent distributions of state, and thence to steady-state distributions; moreover, these steady-state distributions often exhibit an invariance with regard to distributions of service in the network. We consider closed systems in which a fixed and finite number of customers circulate through the network and no external arrivals or departures are permitted, and open systems in which customers originate from an external source according to a Poisson process, possibly non-homogeneous, and each customer eventually leaves the system.