Book contents
- Frontmatter
- Contents
- List of contributors
- Preface
- Bibliography of J. F. C. Kingman
- 1 A fragment of autobiography, 1957–1967
- 2 More uses of exchangeability: representations of complex random structures
- 3 Perfect simulation using dominated coupling from the past with application to area-interaction point processes and wavelet thresholding
- 4 Assessing molecular variability in cancer genomes
- 5 Branching out
- 6 Kingman, category and combinatorics
- 7 Long-range dependence in a Cox process directed by an alternating renewal process
- 8 Kernel methods and minimum contrast estimators for empirical deconvolution
- 9 The coalescent and its descendants
- 10 Kingman and mathematical population genetics
- 11 Characterizations of exchangeable partitions and random discrete distributions by deletion properties
- 12 Applying coupon-collecting theory to computer-aided assessments
- 13 Colouring and breaking sticks: random distributions and heterogeneous clustering
- 14 The associated random walk and martingales in random walks with stationary increments
- 15 Diffusion processes and coalescent trees
- 16 Three problems for the clairvoyant demon
- 17 Homogenization for advection-diffusion in a perforated domain
- 18 Heavy traffic on a controlled motorway
- 19 Coupling time distribution asymptotics for some couplings of the Lévy stochastic area
- 20 Queueing with neighbours
- 21 Optimal information feed
- 22 A dynamical-system picture of a simple branching-process phase transition
- Index
19 - Coupling time distribution asymptotics for some couplings of the Lévy stochastic area
Published online by Cambridge University Press: 07 September 2011
- Frontmatter
- Contents
- List of contributors
- Preface
- Bibliography of J. F. C. Kingman
- 1 A fragment of autobiography, 1957–1967
- 2 More uses of exchangeability: representations of complex random structures
- 3 Perfect simulation using dominated coupling from the past with application to area-interaction point processes and wavelet thresholding
- 4 Assessing molecular variability in cancer genomes
- 5 Branching out
- 6 Kingman, category and combinatorics
- 7 Long-range dependence in a Cox process directed by an alternating renewal process
- 8 Kernel methods and minimum contrast estimators for empirical deconvolution
- 9 The coalescent and its descendants
- 10 Kingman and mathematical population genetics
- 11 Characterizations of exchangeable partitions and random discrete distributions by deletion properties
- 12 Applying coupon-collecting theory to computer-aided assessments
- 13 Colouring and breaking sticks: random distributions and heterogeneous clustering
- 14 The associated random walk and martingales in random walks with stationary increments
- 15 Diffusion processes and coalescent trees
- 16 Three problems for the clairvoyant demon
- 17 Homogenization for advection-diffusion in a perforated domain
- 18 Heavy traffic on a controlled motorway
- 19 Coupling time distribution asymptotics for some couplings of the Lévy stochastic area
- 20 Queueing with neighbours
- 21 Optimal information feed
- 22 A dynamical-system picture of a simple branching-process phase transition
- Index
Summary
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?
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- Information
- Probability and Mathematical GeneticsPapers in Honour of Sir John Kingman, pp. 446 - 463Publisher: Cambridge University PressPrint publication year: 2010
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