Consider a decision-maker who is in charge of a number of activities. At each of a sequence of decision points in time, he selects—on the basis of their performance—the set of activities to be continued further and allocates his limited resources among them. Activities receiving larger allocations tend to improve their performance, while others receiving smaller allocations tend to deteriorate. We present a controlled random walk model for the progress of these activities. The problem of maximizing the net infinite horizon discounted return is formulated in the framework of Markov decision theory, and existence of optimal strategies established. It is shown that both the optimal selection and allocation strategies exhibit a ‘favoring the leaders' behavior. Finally, explicit solutions to certain special cases are obtained illustrating these results.

The purpose of the present paper is to give a general theory of the quantum mechanical representation of particle populations.

The first part of the paper, Sections 1 to 5, is devoted to a review of mathematical principles of quantum theory, with particular emphasis on the role played by probability concepts, using an approach adapted to the subsequent development of the theory of particle populations. This approach, which goes back in its essentials to von Neumann [20], leans heavily on the subsequent work of Wigner, Mackey, Jauch, Segal, Wightman and many others (see e.g., Mackey [15], Jauch [11], Streater and Wightman [26]). Sections 6 to 9 deal with the representation of finite particle populations: i.e., quantum systems where the total number of particles is an observable. In Section 10 a brief sketch is given of the generalization of the theory to infinite populations where the total number of particles is not an observable, as e.g., in the statistical theory of an infinitely extended gas (see Ruelle [22]). Finally, Section 11 treats some simple examples.

Let be a real-valued, homogeneous, and isotropic random field indexed in . When restricted to those indices with , the Euclidean length of , equal to r (a positive constant), then the random field resides on the surface of a sphere of radius r. Using a modified stratified spherical sampling plan (Brown (1993a)) on the sphere, define to be a realization of the random process and to be the cardinality of . A bootstrap algorithm is presented and conditions for strong uniform consistency of the bootstrap cumulative distribution function of the standardized sample mean, , are given. We illustrate the bootstrap algorithm with global land-area data.

Recent research at Sheffield, which is briefly described below, has been concerned with chain-binomial models, and stochastic path integrals under infective curves.

In a multi-server queueing system in which the customers are of several different types, it is useful to define states which specify the types of customers being served as well as the total number present. Analogies with some problems in statistical mechanics are found fruitful. Certain generating functions are defined in such a way that they satisfy a system of linear equations. Solution of the associated eigenvector problem shows that the steady-state probabilities for states in which all the servers are busy can be represented by a weighted sum of geometric probabilities.

A method originating from statistical mechanics (low-density and high-temperature expansions) is used to prove the existence and uniqueness of a stationary regime for switching networks on finite or infinite graphs. The main assumption is that the message (customer) flows circulating through the network are ‘localized' in the sense that, for any message, the probability of having a long path is rapidly decreasing (and, moreover, a path of a ‘typical' message consists of one line). The switching rule combines message-switching and circuit-switching principles. The stationary regime for the network under consideration may be treated as a ‘small perturbation' of the ‘idealized' regime in the totally decoupled network where all the messages have single line paths.

We construct an immigration-branching process from an inhomogeneous Poisson process, a parameter-dependent probability distribution of populations and a Markov branching process with homogeneous transition function. The set of types is arbitrary, and the parameter is allowed to be discrete or continuous. Assuming a supercritical branching part with primitive first moments and finite second moments, we prove propositions on the mean square convergence and the almost sure convergence of normalized averaging processes associated with the immigration-branching process.

This paper is concerned with everywhere local behaviour of certain classes of random processes which have stationary Gaussian increments. It is shown that for two classes of processes almost all the sample functions have the following property. The supremum of the increments in the neighbourhood of a point is everywhere of larger order than the standard deviation. For a third class of processes it is shown that the supremum is at least of the same order as the standard deviation.