Univariate approximations in the infinite occupancy scheme

The paper concerns the classical occupancy scheme with infinitely many boxes. We establish approximations to the distributions of the number of occupied boxes, and of the number of boxes containing exactly r balls, within the family of translated Poisson distributions. These are shown to be of ideal asymptotic order, with respect both to total variation distance and to the approximation of point probabilities. The proof is probabilistic, making use of a translated Poisson approximation theorem of R\"ollin (2005).Comment: 20 page

Translated Poisson approximation to equilibrium distributions of Markov population processes

The paper is concerned with the equilibrium distributions of continuous-time density dependent Markov processes on the integers. These distributions are known typically to be approximately normal, and the approximation error, as measured in Kolmogorov distance, is of the smallest order that is compatible with their having integer support. Here, an approximation in the much stronger total variation norm is established, without any loss in the asymptotic order of accuracy; the approximating distribution is a translated Poisson distribution having the same variance and (almost) the same mean. Our arguments are based on the Stein-Chen method and Dynkin's formula.Comment: 18 page

A central limit theorem for the gossip process

The Aldous gossip process represents the dissemination of information in geographical space as a process of locally deterministic spread, augmented by random long range transmissions. Starting from a single initially informed individual, the proportion of individuals informed follows an almost deterministic path, but for a random time shift, caused by the stochastic behaviour in the very early stages of development. In this paper, it is shown that, even with the extra information available after a substantial development time, this broad description remains accurate to first order. However, the precision of the prediction is now much greater, and the random time shift is shown to have an approximately normal distribution, with mean and variance that can be computed from the current state of the process

Central limit theorems in the configuration model

We prove a general normal approximation theorem for local graph statistics in the configuration model, together with an explicit bound on the error in the approximation with respect to the Wasserstein metric. Such statistics take the form $T := \sum_{v \in V} H_v$, where $V$ is the vertex set, and $H_v$ depends on a neighbourhood in the graph around $v$ of size at most $\ell$. The error bound is expressed in terms of $\ell$, $|V|$, an almost sure bound on $H_v$, the maximum vertex degree $d_{\max}$ and the variance of $T$. Under suitable assumptions on the convergence of the empirical degree distributions to a limiting distribution, we deduce that the size of the giant component in the configuration model has asymptotically Gaussian fluctuations.Comment: minor change

Asymptotic behaviour of gossip processes and small world networks

Both small world models of random networks with occasional long range connections and gossip processes with occasional long range transmission of information have similar characteristic behaviour. The long range elements appreciably reduce the effective distances, measured in space or in time, between pairs of typical points. In this paper, we show that their common behaviour can be interpreted as a product of the locally branching nature of the models. In particular, it is shown that both typical distances between points and the proportion of space that can be reached within a given distance or time can be approximated by formulae involving the limit random variable of the branching process.Comment: 30 page

Assessing molecular variability in cancer genomes

The dynamics of tumour evolution are not well understood. In this paper we provide a statistical framework for evaluating the molecular variation observed in different parts of a colorectal tumour. A multi-sample version of the Ewens Sampling Formula forms the basis for our modelling of the data, and we provide a simulation procedure for use in obtaining reference distributions for the statistics of interest. We also describe the large-sample asymptotics of the joint distributions of the variation observed in different parts of the tumour. While actual data should be evaluated with reference to the simulation procedure, the asymptotics serve to provide theoretical guidelines, for instance with reference to the choice of possible statistics.Comment: 22 pages, 1 figure. Chapter 4 of "Probability and Mathematical Genetics: Papers in Honour of Sir John Kingman" (Editors N.H. Bingham and C.M. Goldie), Cambridge University Press, 201