Probabilistic symmetries and invariance principles / Olav Kallenberg.

Format:

Book

Author/Creator:

Kallenberg, Olav.

Contributor:

Bernard W. Freeman Book Fund.

Series:

Probability and its applications (Springer-Verlag)

Probability and its applications

Language:

English

Subjects (All):

Probabilities.

Symmetry (Physics).

Physical Description:

xi, 510 pages ; 25 cm.

Place of Publication:

New York : Springer, [2005]

Summary:

This is the first comprehensive treatment of the three basic symmetries of probability theory-contractability, exchangeability, and rotatability-defined as invariance in distribution under contractions, permutations, and rotations. Originating with the pioneering work of de Finetti from the 1930's, the theory has evolved into a unique body of deep, beautiful, and often surprising results. These results comprise the basic representations and invariance properties in one and several dimensions and exhibit some unexpected links between the various symmetries as well as to many other areas of modern probability. Most chapters require only some basic, graduate level probability theory, and should be accessible to any serious researchers and graduate students in probability and statistics. Parts of the book may also be of interest to pure and applied mathematicians in other areas. The exposition is formally self-contained, with detailed references provided for any deeper facts from real analysis or probability used in the book.

Contents:

1 The Basic Symmetries 24

1.1 Infinite sequences 24

1.2 Finite sequences 30

1.3 Continuous-time symmetries 35

1.4 Infinite-interval processes 43

1.5 Measures on a finite interval 46

1.6 Simple or diffuse random measures 52

1.7 Rotations and L[superscript p]-symmetries 57

1.8 Miscellaneous complements 66

2 Conditioning and Martingales 69

2.1 Contractable sequences 69

2.2 Continuous-time symmetries 75

2.3 Semi-martingale criteria 83

2.4 Further criteria and representation 90

2.5 Norm relations and regularity 95

2.6 Path properties 103

2.7 Palm measure invariance 111

3 Convergence and Approximation 125

3.1 Discrete-time case 125

3.2 Random measures 129

3.3 Exchangeable processes 136

3.4 Approximation and representation 143

3.5 Restriction and extension 149

3.6 Coupling and path properties 155

3.7 Sub-sequence principles 162

4 Predictable Sampling and Mapping 169

4.1 Skipping and sampling 169

4.2 Gauss and Poisson reduction 173

4.3 Predictable mapping 176

4.4 Predictable contraction 185

4.5 Brownian and stable invariance 189

4.6 Mapping of optional times 200

5 Decoupling Identities 209

5.1 Integrability and norm estimates 209

5.2 Exchangeable sums 216

5.3 Martingale representations 223

5.4 Exchangeable integrals 229

5.5 Levy integrals 233

5.6 Contractable sums and integrals 240

5.7 Predictable sampling revisited 249

6 Homogeneity and Reflections 255

6.1 Symmetries and dichotomies 255

6.2 Local homogeneity 261

6.3 Reflection invariance 268

6.4 Local time and intensity 271

6.5 Exponential and uniform sampling 279

6.6 Hitting points and intervals 285

6.7 Markov properties 290

6.8 Homogeneity and independence 294

7 Symmetric Arrays 300

7.1 Notation and basic symmetries 300

7.2 Coupling, extension, and independence 305

7.3 Coding and inversion 310

7.4 Contractable arrays 318

7.5 Exchangeable arrays 325

7.6 Equivalence criteria 328

7.7 Conditional distributions 335

7.8 Symmetric partitions 342

8 Multi-variate Rotations 350

8.1 Rotational symmetries 350

8.2 Gaussian and rotatable processes 356

8.3 Functionals on a product space 359

8.4 Preliminaries for rotatable arrays 363

8.5 Separately rotatable arrays and functionals 370

8.6 Jointly rotatable functionals 378

8.7 Jointly rotatable arrays 384

8.8 Separately exchangeable sheets 391

8.9 Jointly exchangeable or contractable sheets 395

9 Symmetric Measures in the Plane 401

9.1 Notions of invariance 401

9.2 General prerequisites 405

9.3 Symmetries on a square 411

9.4 Symmetry on a strip 416

9.5 Technical preparation 422

9.6 Symmetries on a quadrant 432

A1 Decomposition and selection 440

A2 Weak convergence 445

A3 Multiple stochastic integrals 449

A4 Complete monotonicity 457

A5 Palm and Papangelou kernels 459.

Notes:

Includes bibliographical references (pages [477]-495) and indexes.

Local Notes:

Acquired for the Penn Libraries with assistance from the Bernard W. Freeman Book Fund.

ISBN:

0387251154

OCLC:

60740995

Publisher Number:

9780387251158