John Frank Charles Kingman was born on 28th August 1939, a few days before the outbreak of World War II. This Festschrift is in honour of his seventieth birthday.

John Kingman was born in Beckenham, Kent, the son of the scientist Dr F. E. T. Kingman FRSC and the grandson of a coalminer. He was brought up in north London, where he attended Christ's College, Finchley. He was an undergraduate at Cambridge, where at age 19 at the end of his second year he took a First in Part II of the Mathematical Tripos, following it with a Distinction in the graduate-level Part III a year later, for his degree. He began postgraduate work as a research student under Peter Whittle, but transferred to David Kendall in Oxford when Peter left for Manchester in 1961, returning to Cambridge when Kendall became the first Professor of Mathematical Statistics there in 1962.

John's early work was on queueing theory, a subject he had worked on with Whittle, but was also an interest of Kendall's. His lifelong interest in mathematical genetics also dates back to this time (1961). His next major interest was in Markov chains, and in a related matter—what happens to Feller's theory of recurrent events in continuous time. His first work here dates from 1962, and led to his landmark 1964 paper on regenerative phenomena, where we meet (Kingman) p-functions.

Abstract

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.

AMS subject classification (MSC2010) 92D20; 92D15, 92C50, 60C05, 62E17

Introduction

Cancers are thought to develop as clonal expansions from a single transformed, ancestral cell. Large-scale sequencing studies have shown that cancer genomes contain somatic mutations occurring in many genes; cf. Greenman et al., Sjöblom et al., Shah et al. Many of these mutations are thought to be passenger mutations (those that are not driving the behaviour of the tumour), and some are pathogenic driver mutations that influence the growth of the tumour. The dynamics of tumour evolution are not well understood, in part because serial observation of tumour growth in humans is not possible.

In an attempt to better understand tumour growth and structure, a number of evolutionary approaches have been described.

Abstract

We exhibit some explicit co-adapted couplings for n-dimensional Brownian motion and all its Lévy stochastic areas. In the two-dimensional case we show how to derive exact asymptotics for the coupling time under various mixed coupling strategies, using Dufresne's formula for the distribution of exponential functionals of Brownian motion. This yields quantitative asymptotics for the distributions of random times required for certain simultaneous couplings of stochastic area and Brownian motion. The approach also applies to higher dimensions, but will then lead to upper and lower bounds rather than exact asymptotics.

Keywords Brownian motion, co-adapted coupling, coupling time distribution, Dufresne formula, exponential functional of Brownian motion, Kolmogorov diffusion, Lévy stochastic area, maximal coupling, Morse– Thue sequence, non-co-adapted coupling, reflection coupling, rotation coupling, stochastic differential, synchronous coupling

AMS subject classification (MSC2010) 60J65, 60H10

Introduction

It is a pleasure to present this paper as a homage to my DPhil supervisor John Kingman, in grateful acknowledgement of the formative period which I spent as his research student at Oxford, which launched me into a deeply satisfying exploration of the world of mathematical research. It seems fitting in this paper to present an overview of a particular aspect of probabilistic coupling theory which has fascinated me for a considerable time; given that one can couple two copies of a random process, when can one in addition couple other associated functionals of the processes? How far can one go?

Abstract

Unlimited access to a motorway network can, in overloaded conditions, cause a loss of capacity. Ramp metering (signals on slip roads to control access to the motorway) can help avoid this loss of capacity. The design of ramp metering strategies has several features in common with the design of access control mechanisms in communication networks.

Inspired by models and rate control mechanisms developed for Internet congestion control, we propose a Brownian network model as an approximate model for a controlled motorway and consider it operating under a proportionally fair ramp metering policy. We present an analysis of the performance of this model.

AMS subject classification (MSC2010) 90B15, 90B20, 60K30

Introduction

The study of heavy traffic in queueing systems began in the 1960s, with three pioneering papers by Kingman [26, 27, 28]. These papers, and the early work of Prohorov [35], Borovkov [5, 6] and Iglehart [20], concerned a single resource. Since then there has been significant interest in networks of resources, with major advances by Harrison and Reiman [19], Reiman [37], Williams [43] and Bramson [7]. For discussions, further references and overviews of the very extensive literature on heavy traffic for networks, Williams [42], Bramson and Dai [8], Harrison [17, 18] and Whitt [41] are recommended.

Research in this area is motivated in part by the need to understand and control the behaviour of communications, manufacturing and service networks, and thus to improve their design and performance.

Abstract

This paper proves certain results from the ‘appetizer for non-linear Wiener–Hopf theory’ [5]. Like that paper, it considers only the simplest possible case in which the underlying Markov process is a two-state Markov chain. Key generating functions provide solutions of a simple two-dimensional dynamical system, and the main interest is in the way in which Probability Theory and ODE theory complement each other. No knowledge of either ODE theory or Wiener–Hopf theory is assumed. Theorem 1.1 describes one aspect of a phase transition which is more strikingly conveyed by Figures 4.1 and 4.2.

AMS subject classification (MSC2010) 60J80, 34A34

Introduction

This paper is a development of something I mentioned briefly in talks I gave at Bristol, when John Kingman was in the audience, and at the Waves conference in honour of John Toland at Bath. I thanked both John K and John T for splendid mathematics and for their wisdom and kindness.

The main point of the paper is to prove Theorem 1.1 and related results in a way which emphasizes connections with a simple dynamical system. The phase transition between Figures 4.1 and 4.2 looks more dramatic than the famous 1-dimensional result we teach to all students.

The model studied here is a special case of the model introduced in Williams [5]. I called that paper, which contained no proofs, an ‘appetizer’; but before writing a fuller version, I became caught up in Jonathan Warren's enthusiasm for the relevance of complex dynamical systems (in ℂ2).

Abstract

We survey classical kernel methods for providing nonparametric solutions to problems involving measurement error. In particular we outline kernel-based methodology in this setting, and discuss its basic properties. Then we point to close connections that exist between kernel methods and much newer approaches based on minimum contrast techniques. The connections are through use of the sinc kernel for kernel-based inference. This ‘infinite order’ kernel is not often used explicitly for kernel-based deconvolution, although it has received attention in more conventional problems where measurement error is not an issue. We show that in a comparison between kernel methods for density deconvolution, and their counterparts based on minimum contrast, the two approaches give identical results on a grid which becomes increasingly fine as the bandwidth decreases. In consequence, the main numerical differences between these two techniques are arguably the result of different approaches to choosing smoothing prameters.

Keywords bandwidth, inverse problems, kernel estimators, local linear methods, local polynomial methods, minimum contrast methods, non-parametric curve estimation, nonparametric density estimation, non-parametric regression, penalised contrast methods, rate of convergence, sinc kernel, statistical smoothing

AMS subject classification (MSC2010) 62G08, 62G051

Introduction

Summary

Our aim in this paper is to give a brief survey of kernel methods for solving problems involving measurement error, for example problems involving density deconvolution or regression with errors in variables, and to relate these ‘classical’ methods (they are now about twenty years old) to new approaches based on minimum contrast methods.

Abstract

We review old and new uses of exchangeability, emphasizing the general theme of exchangeable representations of complex random structures. Illustrations of this theme include processes of stochastic coalescence and fragmentation; continuum random trees; second-order limits of distances in random graphs; isometry classes of metric spaces with probability measures; limits of dense random graphs; and more sophisticated uses in finitary combinatorics.

AMS subject classification (MSC2010) 60G09, 60C05, 05C80

Introduction

Kingman's write-up of his 1977 Wald Lectures drew attention to the subject of exchangeability, and further indication of the topics of interest around that time can be seen in the write-up of my 1983 Saint-Flour lectures. As with any mathematical subject, one might expect some topics subsequently to wither, some to blossom and new topics to emerge unanticipated. This Festschrift paper aims, in informal lecture style,

1. (a) to recall the state of affairs 25 years ago (sections 2.1–2.3, 3.1);

2. (b) to briefly describe three directions of subsequent development that have recently featured prominently in monographs (sections 2.4, 3.1–3.2);

3. (c) to describe 3 recent rediscoveries, motivated by new theoretical topics outside mainstream mathematical probability, of the theory of representations of partially exchangeable arrays (sections 2.5, 5.1–5.2);

4. (d) to emphasize a general program that has interested me for 20 years. It doesn't have a standard name, but let me here call it exchangeable representations of complex random structures (section 4).