Mark Westcott

A self-correcting point process

Suppose a point process is attempting to operate as closely as possible to a deterministic rate [rho], in the sense of aiming to produce [rho]t points during the interval (0,t] for all t. This can be modelled by making the instantaneous rate of t of the process a suitable function of n-[rho]t, n being the number of points in [0, t]. This paper studies such a self-correcting point process in two cases: when the point process is Markovian and the rate function is very general, and when the point process is arbitrary and the rate function is exponential. In each case it is shown that as t-->[infinity] the mean number of points occuring in (0, t] is [rho]t+O(1) while the variance is bounded further, in the Markov case all the absolute central moments are bounded. An application to the outputs of stationary D/M/s queues is given.

Clustering estimates for spatial point distributions with unstable potentials

SummarySeveral authors have tried to model highly clustered point patterns by using Gibbs distributions with attractive potentials. Some of these potentials violate a stability condition well known in statistical mechanics. We show that such potentials produce patterns which are much more tightly clustered than those considered by the authors. More generally, our estimates provide a useful test for rejecting unsuitable potentials in models for given patterns. We also use instability arguments to reject related approximations and simulations.

On the Distributions of Scan Statistics

Abstract This article considers the distribution of the scan statistic over some interval of the real line, both for a fixed number of independent uniform points and for a Poisson process. It provides a new series method of calculating this distribution in the Poisson case and a simple self-contained recursive approximation in the fixed case. These results work best at low densities, which is precisely the case when the closed formulas are computationally impractical. The results are compared with values tabulated by Neff and Naus (1980) and with a recent approximation due to Naus (1982). There is also an explanation of why Nauss approximation is good in the Poisson case and a derivation of recursive bounds in the fixed case.

Measures of Societal Risk and Their Potential Use in Civil Aviation

This article seeks to clarify the conceptual foundations of measures of societal risk, to investigate how such measures may be used validly in commonly encountered policy contexts, and to explore the application of these measures in the field of civil aviation. The article begins by examining standard measures of societal and individual risk (SR and IR), with attention given to ethical as well as analytical considerations. A comprehensive technical analysis of SR is provided, encompassing scalar risk measures, barrier functions, and a utility-based formulation, and clarifications are offered with respect to the treatment of SR in recent publications. The policy context for SR measures is shown to be critically important, and an extension to a hierarchical setting is developed. The prospects for applying SR to civil aviation are then considered, and some technical and conceptual issues are identified. SR appears to be a useful analytical tool in this context, provided that careful attention is given to these issues.

Negative skewness and negative correlations in spatial competition models

We consider a population of identical biological entities that compete with each other for space and derive, under rather weak assumptions, some statistical properties of the sizes (e.g. biomasses) of the individuals. It is proved, for example, that, during the early stages of competition, their size distribution becomes negatively skewed and the sizes of neighbouring individuals become negatively correlated. The results are proved for a regular periodic array of centres of the individuals where they can be compared with data on plantations.

Random walks on a lattice

This paper discusses the mean-square displacement for a random walk on a two-dimensional lattice, whose transitions to nearest-neighbor sites are symmetric in the horizontal and vertical directions and depend on the column currently occupied. Under a uniform density condition for the step probabilities it is shown that the horizontal mean-square displacement aftern steps is asymptotically proportional ton, and independent of the particular column configuration. The model generalizes that of Seshadri, Lindenberg, and Shuler and the arguments are essentially probabilistic.

Stationary states of crystal growth in three dimensions

A new Markov process describing crystal growth in three dimensions is introduced. States of the process are configurations of the crystal surface, which has a terrace-edge-kink structure. The states are continuous along edges but discrete across edges, in accordance with the very different rates for the two types of captures of particles. Stationary distributions, describing steady crystal growth, are found in general. To our knowledge, these are the first examples of stationary distributions for layered crystal growth in three dimensions. The steady growth rate and other quantities are obtained explicitly for two interacting edges. For many interacting edges, growth behavior is determined (a) in various asymptotic regimes including thermodynamic limits, (b) via simulations, and (c) using series (cluster) expansions in the slope of the surface, the first three coefficients being computed. The theoretical growth rates show a marked dependence on surface dimensions. This may contribute to the size dependence and dispersion in the observed growth rate of small crystals.

The asymptotic behavior of a random walk on a dual-medium lattice

This paper considers the asymptotic distribution for the horizontal displacement of a random walk in a medium represented by a two-dimensional lattice, whose transitions are to nearest-neighbor sites, are symmetric in the horizontal and vertical directions, and depend on the column currently occupied. On either side of a change-point in the medium, the transition probabilities are assumed to obey an asymptotic density condition. The displacement, when suitably normalized, converges to a diffusion process of oscillating Brownian motion type. Various special cases are discussed.

Predicting Fiber Contact in a Three-Dimensional Model of Paper

We present a new methodology for the mathematical analysis of 3D paper structure and apply it to a model that can be simulated on a computer. We rigorously derive upper and lower bounds for fiber contact areas, and derive an approximation that is very close to values calculated from the simulations. The method involves the stochastic geometry and combinatorics of large numbers of randomly located sets, which quantify the interactions between fibers. The main calculation involves a sum resembling a partition function with many-body interactions of all orders.

On the stability of crystal growth

We investigate properties of solid-on-solid models for crystal growth, involving general microscopic rates of capture of atoms by the crystal surface and of escape of atoms. The rates in this Markov process influence the stability of the growing surface. We prove, for various different ranges of the rate parameters, stability (i.e., ergodicity) and instability (i.e., nullity) of the growth process. Symmetry properties of the process, such as reversibility, dynamic reversibility, and reflection invariance, are proved or disproved under various conditions. We give a measure of surface smoothness that distinguishes between stable and unstable growth.