A. D. Barbour

Small worlds

Small world models are networks consisting of many local links and fewer long range ‘shortcuts’. In this paper, we consider some particular instances, and rigorously investigate the distribution of their inter-point network distances. Our results are framed in terms of approximations, whose accuracy increases with the size of the network. We also give some insight into how the reduction in typical inter-point distances occasioned by the presence of shortcuts is related to the dimension of the underlying space.

An introduction to Stein's method

A common theme in probability theory is the approximation of complicated probability distributions by simpler ones, the central limit theorem being a classical example. Steins method is a tool which makes this possible in a wide variety of situations. Traditional approaches, for example using Fourier analysis, become awkward to carry through in situations in which dependence plays an important part, whereas Steins method can often still be applied to great effect. In addition, the method delivers estimates for the error in the approximation, and not just a proof of convergence. Nor is there in principle any restriction on the distribution to be approximated; it can equally well be normal, or Poisson, or that of the whole path of a random process, though the techniques have so far been worked out in much more detail for the classical approximation theorems. This volume of lecture notes provides a detailed introduction to the theory and application of Steins method, in a form suitable for graduate students who want to acquaint themselves with the method. It includes chapters treating normal, Poisson and compound Poisson approximation, approximation by Poisson processes, and approximation by an arbitrary distribution, written by experts in the different fields. The lectures take the reader from the very basics of Steins method to the limits of current knowledge.

Stein's method and poisson process convergence

Steins method of obtaining rates of convergence, well known in normal and Poisson approximation, is considered here in the context of approximation by Poisson point processes, rather than their one-dimensional distributions. A general technique is sketched, whereby the basic ingredients necessary for the application of Steins method may be derived, and this is applied to a simple problem in Poisson point process approximation.

Stein's method for diffusion approximations

SummarySteins method of obtaining distributional approximations is developed in the context of functional approximation by the Wiener process and other Gaussian processes. An appropriate analogue of the one-dimensional Stein equation is derived, and the necessary properties of its solutions are established. The method is applied to the partial sums of stationary sequences and of dissociated arrays, to a process version of the Wald-Wolfowitz theorem and to the empirical distribution function.

Macdonald's model and the transmission of bilharzia.

The paper considers a model for the transmission of bilharzia based on Macdonalds assumptions, in the light of data observed in the field. It is shown, in particular, that the threshold parameter governing whether or not an endemic cycle can be established is closely related to the proportion of infected snails in a community, and that this proportion is normally observed to be rather smaller than is compatible with the model. By considering more sophisticated models, allowing for the latent period of infection in the snails, and also for spatial and seasonal heterogeneity, the effective proportion of infected snails, from the point of view of Macdonalds model, is shown to be rather larger, and expressions are given whereby it can be evaluated from observable quantities. However, for the data from Malirong which are taken as illustration, it is also demonstrated that an even more plausible threshold value is obtained from a simple model incorporating human immunity in addition to the assumptions of Macdonalds model, and that, if this model were reasonable, human immunity would appear to be the most important factor in controlling the level of the disease in Malirong.

A central limit theorem for decomposable random variables with applications to random graphs

Abstract The application of Steins method of obtaining rates of convergence to the normal distribution is illustrated in the context of random graph theory. Problems which exhibit a dissociated structure and problems which do not are considered. Results are obtained for the number of copies of a given graph G in K ( n , p ), for the number of induced copies of G , for the number of isolated trees of order k ≥ 2, for the number of vertices of degree d ≥ 1, and for the number of isolated vertices.

Poisson approximation for some statistics based on exchangeable trials

Steins (1970) method of proving limit theorems for sums of dependent random variables is used to derive Poisson approximations for a class of statistics, constructed from finitely exchangeable random variables. Let be exchangeable random elements of a space and, for I a k -subset of , let X I be a 0–1 function. The statistics studied here are of the form where N is some collection of k -subsets of . An estimate of the total variation distance between the distributions of W and an appropriate Poisson random variable is derived and is used to give conditions sufficient for W to be asymptotically Poisson. Two applications of these results are presented.

Procedures for reliable estimation of viral fitness from time-series data

In order to develop a better understanding of the evolutionary dynamics of HIV drug resistance, it is necessary to quantify accurately the in vivo fitness costs of resistance mutations. However, the reliable estimation of such fitness costs is riddled with both theoretical and experimental difficulties. Experimental fitness assays typically suffer from the shortcoming that they are based on in vitro data. Fitness estimates based on the mathematical analysis of in vivo data, however, are often questionable because the underlying assumptions are not fulfilled. In particular, the assumption that the replication rate of the virus population is constant in time is frequently grossly violated. By extending recent work of Marée and colleagues, we present here a new approach that corrects for time–dependent viral replication in time–series data for growth competition of mutants. This approach allows a reliable estimation of the relative replicative capacity (with confidence intervals) of two competing virus variants growing within the same patient, using longitudinal data for the total plasma virus load, the relative frequency of the two variants and the death rate of infected cells. We assess the accuracy of our method using computer–generated data. An implementation of the developed method is freely accessible on the Web (http://www.eco.ethz.ch/fitness.html).

Epidemics and random graphs

The idea of this note is to point out that the simple random graph G(η,ρ) (Bollobas (1985)) can be used as an internal description for the Reed-Frost.cha~n-binomial epidemic model (Bailey (1975)). This relationship is mutually beneficml: It allows us to deduce new results on each model from old results on the other, and also points to possible extensions of work on each. It also makes clear that the Reed-Frost model, in its contact structure if not its development in time, is, pace Jacquez (1987), one of the simplest and most elegant of epidemic models (see also Lefevre and Picard, 1989).

Asymptotic expansions based on smooth functions in the central limit theorem

SummarySteins method is used to derive asymptotic expansions for expectations of smooth functions of sums of independent random variables, together with Lyapounov estimates of the error in the approximation.