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, comprising the basic representations and invariance properties in one and several dimensions, and exhibiting 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.

This book is about random objects--sequences, processes, arrays, measures, functionals--with interesting symmetry properties.

The Basic Symmetries.- Conditioning and Martingales.- Convergence and Approximation.- Predictable Sampling and Mapping.- Decoupling Identities.- Homogeneity and Reflections.- Symmetric Arrays.- Multi-variate Rotations.- Symmetric Measures in the Plane.