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).

Abstract

A number of tricky problems in probability are discussed, having in common one or more infinite sequences of coin tosses, and a representation as a problem in dependent percolation. Three of these problems are of ‘Winkler’ type, that is, they are challenges for a clairvoyant demon.

Keywords clairvoyant demon, dependent percolation, multiscale analysis, percolation, percolation of words, random walk

AMS subject classification (MSC2010) 60K35, 82B20, 60E15

Introduction

Probability theory has emerged in recent decades as a crossroads where many sub-disciplines of mathematical science meet and interact. Of the many examples within mathematics, we mention (not in order): analysis, partial differential equations, mathematical physics, measure theory, discrete mathematics, theoretical computer science, and number theory. The International Mathematical Union and the Abel Memorial Fund have recently accorded acclaim to probabilists. This process of recognition by others has been too slow, and would have been slower without the efforts of distinguished mathematicians including John Kingman.

JFCK's work looks towards both theory and applications. To single out just two of his theorems: the subadditive ergodic theorem [22, 23] is a piece of mathematical perfection which has also proved rather useful in practice; his ‘coalescent’ [24, 25] is a beautiful piece of probability, now a keystone of mathematical genetics. John is also an inspiring and devoted lecturer, who continued to lecture to undergraduates even as the Bristol Vice-Chancellor, and the Director of the Isaac Newton Institute in Cambridge.

Abstract

Kingman's Theorem on skeleton limits—passing from limits as n → ∞ along nh (n ∈ ℕ) for enough h > 0 to limits as t → ∞ for t ∈ ℝ— is generalized to a Baire/measurable setting via a topological approach. We explore its affinity with a combinatorial theorem due to Kestelman and to Borwein and Ditor, and another due to Bergelson, Hindman and Weiss. As applications, a theory of ‘rational’ skeletons akin to Kingman's integer skeletons, and more appropriate to a measurable setting, is developed, and two combinatorial results in the spirit of van der Waerden's celebrated theorem on arithmetic progressions are given.

Keywords Baire property, bitopology, complete metrizability, density topology, discrete skeleton, essential contiguity, generic property, infinite combinatorics, measurable function, Ramsey theory, Steinhaus theory

AMS subject classification (MSC2010) 26A03

Introduction

The background to the theme of the title is Feller's theory of recurrent events. This goes back to Feller in 1949 [F1], and received its first textbook synthesis in [F2] (see e.g. [GS] for a recent treatment). One is interested in something (‘it’, let us say for now—we can proceed informally here, referring to the above for details) that happens (by default, or by fiat) at time 0, may or may not happen at discrete times n = 1, 2, …, and is such that its happening ‘resets the clock’, so that if one treats this random time as a new time-origin, the subsequent history is a probabilistic replica of the original situation.

It is a great honour to have been given this fine collection, and I am most grateful to the Editors and to my friends who have contributed to it. It is poor thanks to inflict on them this self-centred account, and I have tried to mitigate the offence by limiting it to a particular decade, which begins as a new student arrives at Pembroke College in Cambridge to start a degree course in mathematics, and ends with him as a professor in the new University of Sussex. It was obviously a crucial time in my own mathematical life, but it happens also to have been a very interesting period in the development of probability theory in Britain, in which I was fortunate to play a junior part.

With little reluctance I have restricted myself to the mathematical aspects of my life. The reader will seek in vain for any details of my transition from a schoolboy to a happy husband soon to be a proud father; the latter conditions have proved lasting.

Of course the story goes back long before 1957. It might be said to have started in about 1920, when Charles Kingman, a miner on the small but prosperous North Somerset coalfield, summoned his two sons William and Frank for a serious talk. Charles came of a family of Mendip villagers, his grandfather having been the carter of Ston Easton.

Abstract

Mathematical population genetics is only one of Kingman's many research interests. Nevertheless, his contribution to this field has been crucial, and moved it in several important new directions. Here we outline some aspects of his work which have had a major influence on population genetics theory.

AMS subject classification (MSC2010) 92D25

Introduction

In the early years of the previous century, the main aim of population genetics theory was to validate the Darwinian theory of evolution, using the Mendelian hereditary mechanism as the vehicle for determining how the characteristics of any daughter generation depended on the corresponding characteristics of the parental generation. By the 1960s, however, that aim had been achieved, and the theory largely moved in a new, retrospective and statistical, direction.

This happened because, at that time, data on the genetic constitution of a population, or at least on a sample of individuals from that population, started to become available. What could be inferred about the past history of the population leading to these data? Retrospective questions of this type include: “How do we estimate the time at which mitochondrial Eve, the woman whose mitochondrial DNA is the most recent ancestor of the mitochondrial DNA currently carried in the human population, lived? How can contemporary genetic data be used to track the ‘Out of Africa’ migration? How do we detect signatures of past selective events in our contemporary genomes?” Kingman's famous coalescent theory became a central vehicle for addressing questions such as these.

Abstract

The coalescent revolutionised theoretical population genetics, simplifying, or making possible for the first time, many analyses, proofs, and derivations, and offering crucial insights about the way in which the structure of data in samples from populations depends on the demographic history of the population. However statistical inference under the coalescent model is extremely challenging, effectively because no explicit expressions are available for key sampling probabilities. This led initially to approximation of these probabilities by ingenious application of modern computationally-intensive statistical methods. A key breakthrough occurred when Li and Stephens introduced a different model, similar in spirit to the coalescent, for which efficient calculations are feasible. In turn, the Li and Stephens model has changed statistical inference for the wealth of data now available which documents molecular genetic variation within populations. We briefly review the coalescent and associated measure-valued diffusions, describe the Li and Stephens model, and introduce and apply a generalisation of it for inference of population structure in the presence of linkage disequilibrium.

AMS subject classification (MSC2010) 60J70, 62M05, 92D10

Introduction

John Kingman made a number of incisive and elegant contributions to modelling in the field of genetics, several of which are described elsewhere in this volume. But it is probably the coalescent, or ‘Kingman coalescent’ as it is often known, which has had the greatest impact. Several authors independently developed related ideas around the same time but it was Kingman's description and formulation, together with his proofs of the key robustness results which had the greatest impact in the mathematical genetics community.