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Probability and Mathematical Genetics: Papers in Honour of Sir John Kingman

by: Edited by N. H. Bingham , C. M. Goldie

Probability and Mathematical Genetics: Papers in Honour of Sir John Kingman

by: Edited by N. H. Bingham , C. M. Goldie

ISBN 13:
9780521145770
ISBN 10:
0521145775
Format: Paperback
Copyright: 08/23/2010
Publisher: Cambridge University Press

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Summary

Focussing on the work of Sir John Kingman, one of the world's leading researchers in probability and mathematical genetics, this book touches on the important areas of these subjects in the last 50 years. Leading authorities give a unique insight into a wide range of currently topical problems. Papers in probability concentrate on combinatorial and structural aspects, in particular exchangeability and regeneration. The Kingman coalescent links probability with mathematical genetics and is fundamental to the study of the latter. This has implications across the whole of genomic modelling including the Human Genome Project. Other papers in mathematical population genetics range from statistical aspects including heterogeneous clustering, to the assessment of molecular variability in cancer genomes. Further papers in statistics are concerned with empirical deconvolution, perfect simulation, and wavelets. This book will be warmly received by established experts as well as their students and others interested in the content.

List of contributors	p. xiii
Preface	p. xv
Bibliography of	p. 1
A fragment of autobiography, 1957-1967	p. 17
More uses of exchangeability: representations of complex random structures	p. 35
Introduction	p. 35
Exchangeability	p. 36
Using exchangeability to describe complex structures	p. 43
Construction of, and convergence to, infinite random combinatorial objects	p. 49
Limits of finite deterministic structures	p. 56
Miscellaneous comments	p. 59
Perfect simulation using dominated coupling from the past with application to area-interaction point processes and wavelet thresholding	p. 64
Introduction	p. 64
Perfect simulation	p. 66
Area-interaction processes	p. 71
Nonparametric regression by wavelet thresholding	p. 77
Perfect simulation for wavelet curve estimation	p. 82
Conclusion	p. 88
Assessing molecular variability in cancer genomes	p. 91
Introduction	p. 91
Colorectal cancer data	p. 93
The Ewens sampling formula	p. 95
Analysis of the cancer data	p. 98
Poission approximation	p. 100
Conclusion	p. 110
Branching out	p. 113
Introduction	p. 113
The basic model	p. 115
Spreading out: old results	p. 116
Spreading out: first refinements	p. 119
Spreading out: recent refinements	p. 120
Deterministic theory	p. 122
The multitype case	p. 124
Anomalous spreading	p. 125
Discussion of anomalous spreading	p. 130
Kingman, category and combinatorics	p. 135
Introduction	p. 135
Preliminaries	p. 138
A bitopological Kingman theorem	p. 143
Applications-rational skeletons	p. 151
KBD in van der Waerden style	p. 157
Applications: additive combinatorics	p. 162
Long-range dependence in a Cox process directed by an alternating renewal process	p. 169
Preamble	p. 170
Introduction	p. 170
Stationary renewal and alternating renewal processes	p. 172
Second moments	p. 178
An alternating renewal process not satisfying Condition A	p. 180
Postlude	p. 181
Kernel methods and minimum contrast estimators for empirical deconvolution	p. 185
Introduction	p. 186
Methodology and theory	p. 191
Relationship to minimum contrast methods	p. 195
The coalescent and its descendants	p. 204
Introduction	p. 205
The coalescent and the Fleming-Viot process	p. 206
Inference under the coalescent	p. 213
The Li and Stephens model	p. 215
Application: modelling population structure	p. 221
Kingman and mathematical population genetics	p. 238
Introduction	p. 238
Background	p. 239
Putting it together	p. 244
Robustness	p. 247
A convergence result	p. 248
Partition structures	p. 249
'Age' properties and the GEM distribution	p. 251
The coalescent	p. 256
Other matters	p. 260
Characterizations of exchangeable partitions and random discrete distributions by deletion properties	p. 264
Introduction	p. 265
Partition structures	p. 266
Partially exchangeable partitions	p. 274
Exchangeable partitions	p. 278
The deletion property without the regularity condition	p. 285
Regeneration and ¿-deletion	p. 286
Applying coupon-collecting theory to computer-aided assessments	p. 299
Introduction	p. 299
Coupon collecting	p. 300
How many tests?	p. 302
Asymptotics	p. 303
Proofs for §4	p. 305
Numerical results	p. 314
Discussion	p. 315
Colouring and breaking sticks: random distributions and heterogeneous clustering	p. 319
Introduction	p. 319
Mixture models and the Dirichlet process	p. 320
Applications and generalisations	p. 325
Pólya urn schemes and MCMC samplers	p. 329
A coloured Dirichlet process	p. 333
The associated random walk and martingales in random walks with stationary increments	p. 345
Introduction and definition	p. 345
Three examples	p. 349
Some remarks on duality and asymptotic independence	p. 355
Diffusion processes and coalescent trees	p. 358
Introduction	p. 359
A coalescent dual process	p. 361
Processes with beta stationary distributions and Jacobi polynomial eigenfunctions	p. 368
Subordinated Jacobi diffusion processes	p. 371
Subordinated coalescent process	p. 375
Three problems for the clairvoyant demon	p. 380
Introduction	p. 380
Site percolation	p. 381
Clairvoyant scheduling	p. 383
Clairvoyant compatibility	p. 384
Clairvoyant embedding	p. 385
Dependent percolation	p. 390
Percolation of words	p. 393
Homogenization for advection-diffusion in a perforated domain	p. 397
Introduction	p. 398
Review of homogenization for diffusion with periodic drift	p. 400
Existence of a volume growth rate for a diffusion sausage with periodic drift	p. 402
Estimates for the diffusion sausage	p. 403
Asymptotics of the growth rate for small and large cross-sections	p. 405
Homogenization of the advection-diffusion equation in a perforated domain	p. 408
The case of diffusivity ¿²I	p. 410
Monte Carlo computation of the asymptotic growth rate	p. 411
Heavy traffic on a controlled motorway	p. 416
Introduction	p. 416
A single queue	p. 418
A model of Internet congestion	p. 422
A Brownian network model	p. 426
A model of a controlled motorway	p. 430
Route choices	p. 438
Concluding remarks	p. 442
Coupling time distribution asymptotics for some couplings of the Lévy stochastic area	p. 446
Different kinds of couplings	p. 448
Reflection coupling	p. 450
Coupling more than one feature of the process	p. 451
Conclusion	p. 461
Queueing with neighbours	p. 464
Introduction	p. 464
Results	p. 467
Asymmetric interaction	p. 470
Symmetric interaction	p. 475
Appendix	p. 480
Optimal information feed	p. 483
Interrogation, transmission and coding	p. 483
A tractable infinite-horizon case	p. 486
A dynamical-system picture of a simple branching-process phase transition	p. 491
Introduction	p. 491
Wiener-Hopferization	p. 493
How does ODE theory see the phase transition?	p. 496
Proof of Theorem 1.1 and more	p. 499
Index	p. 509
Table of Contents provided by Ingram. All Rights Reserved.

Amazon no longer offers textbook rentals. We do!

Probability and Mathematical Genetics: Papers in Honour of Sir John Kingman

Probability and Mathematical Genetics: Papers in Honour of Sir John Kingman

9780521145770

0521145775

Summary

Table of Contents

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