Probability and Its Applications

By: Olav Kallenberg

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Hardcover | 27 July 2005

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524 Pages

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23.5 x 15.88 x 2.54

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This book is about random objects-sequences, processes, arrays, measures, functionals-with interesting symmetry properties. Here symmetry should beunderstoodinthebroadsenseofinvarianceunderafamily(notnecessarily a group) of measurable transformations. To be precise, it is not the random objects themselves but rather their distributions that are assumed to be symmetric. Though many probabilistic symmetries are conceivable and have been considered in various contexts, four of them-stationarity, contractability, exchangeability, and rotatability-stand out as especially interesting and - portant in several ways: Their study leads to some deep structural theorems of great beauty and signi?cance, they are intimately related to some basic areasofmodernprobabilitytheory, andtheyaremutuallyconnectedthrough a variety of basic relationships. The mentioned symmetries may be de?ned as invariance in distribution under shifts, contractions, permutations, and rotations. Stationarity being a familiar classical topic, treated extensively in many standard textbooks and monographs, most of our attention will be focused on the remaining three basic symmetries. The study of general probabilistic symmetries essentially originated with the work of de Finetti (1929-30), who proved by elementary means (no - vanced tools being yet available) the celebrated theorem named after him- the fact that every in?nite sequence of exchangeable events is mixed i.i.d.

Industry Reviews

From the reviews of the first edition:

"The starting point of this book ... is de Finetti's famous theorem characterizing sequences of exchangeable random variables as mixtures of i.i.d. random variables. ... The monograph provides a tight approach to this subject based on modern mathematical tools ... and presents de Finetti's theorem in a new framework ... . the monograph under review presents ... a profound and complete treatment of the topic. The reader interested in this field will find a comprehensive investigation leading to the actual frontiers of research." (Wilfried Hazod, Mathematical Reviews, Issue 2006 i)

"This elegantly written, graduate level monograph provides a comprehensive coverage of probabilistic symmetries. ... This book has an extensive bibliography and valuable, detailed historical notes that place the work in context and provide links to areas of application. The text conveys the author's enthusiasm for the subject and his deep understanding of the key structures gained over 30 years of active research into the subject." (Neville Weber, Zentralblatt MATH, Vol. 1084, 2006)

"The author achieves a remarkable level of clarity, economy, and accessibility ... with enviable skill and judgement. ... The book may thus be used as much as read in the future. ... A reader interested in exchangeability will find results a-plenty in this book ... . As a whole, this is a grand conception, superbly realized." (Charles M. Goldie, Journal of the American Statistical Association, Vol. 102 (479), 2007)

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You Can Find This Book In

Non-Fiction Mathematics Probability & Statistics Applied Mathematics Stochastics

Preface	p. v
Introduction	p. 1
The Basic Symmetries	p. 24
Infinite sequences	p. 24
Finite sequences	p. 30
Continuous-time symmetries	p. 35
Infinite-interval processes	p. 43
Measures on a finite interval	p. 46
Simple or diffuse random measures	p. 52
Rotations and L[superscript p]-symmetries	p. 57
Miscellaneous complements	p. 66
Conditioning and Martingales	p. 69
Contractable sequences	p. 69
Continuous-time symmetries	p. 75
Semi-martingale criteria	p. 83
Further criteria and representation	p. 90
Norm relations and regularity	p. 95
Path properties	p. 103
Palm measure invariance	p. 111
Convergence and Approximation	p. 125
Discrete-time case	p. 125
Random measures	p. 129
Exchangeable processes	p. 136
Approximation and representation	p. 143
Restriction and extension	p. 149
Coupling and path properties	p. 155
Sub-sequence principles	p. 162
Predictable Sampling and Mapping	p. 169
Skipping and sampling	p. 169
Gauss and Poisson reduction	p. 173
Predictable mapping	p. 176
Predictable contraction	p. 185
Brownian and stable invariance	p. 189
Mapping of optional times	p. 200
Decoupling Identities	p. 209
Integrability and norm estimates	p. 209
Exchangeable sums	p. 216
Martingale representations	p. 223
Exchangeable integrals	p. 229
Levy integrals	p. 233
Contractable sums and integrals	p. 240
Predictable sampling revisited	p. 249
Homogeneity and Reflections	p. 255
Symmetries and dichotomies	p. 255
Local homogeneity	p. 261
Reflection invariance	p. 268
Local time and intensity	p. 271
Exponential and uniform sampling	p. 279
Hitting points and intervals	p. 285
Markov properties	p. 290
Homogeneity and independence	p. 294
Symmetric Arrays	p. 300
Notation and basic symmetries	p. 300
Coupling, extension, and independence	p. 305
Coding and inversion	p. 310
Contractable arrays	p. 318
Exchangeable arrays	p. 325
Equivalence criteria	p. 328
Conditional distributions	p. 335
Symmetric partitions	p. 342
Multi-variate Rotations	p. 350
Rotational symmetries	p. 350
Gaussian and rotatable processes	p. 356
Functionals on a product space	p. 359
Preliminaries for rotatable arrays	p. 363
Separately rotatable arrays and functionals	p. 370
Jointly rotatable functionals	p. 378
Jointly rotatable arrays	p. 384
Separately exchangeable sheets	p. 391
Jointly exchangeable or contractable sheets	p. 395
Symmetric Measures in the Plane	p. 401
Notions of invariance	p. 401
General prerequisites	p. 405
Symmetries on a square	p. 411
Symmetry on a strip	p. 416
Technical preparation	p. 422
Symmetries on a quadrant	p. 432
Appendices	p. 440
Decomposition and selection	p. 440
Weak convergence	p. 445
Multiple stochastic integrals	p. 449
Complete monotonicity	p. 457
Palm and Papangelou kernels	p. 459
Historical and Bibliographical Notes	p. 464
Bibliography	p. 477
Author Index	p. 497
Subject Index	p. 501
Symbol Index	p. 509
Table of Contents provided by Ingram. All Rights Reserved.

Probability and Its Applications

At a Glance

Hardcover

Industry Reviews

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Defective items

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