Probabilistic symmetries and invariance principles.

*(English)*Zbl 1084.60003
Probability and Its Applications. New York, NY: Springer (ISBN 0-387-25115-4/hbk; 978-1-4419-2042-3/pbk; 978-0-387-28861-1/ebook). xi, 510 p. (2005).

This elegantly written, graduate level monograph provides a comprehensive coverage of probabilistic symmetries. The book is about random objects whose distributions are invariant under a family of measurable transformations. The four key symmetries considered are stationarity, contractability (or spreading invariance), exchangeability, and rotatability. While most results are known there are some new theorems and novel proofs of well known results. The author accurately describes the work as a ‘reasonably complete theory involving the basic notions of distributional symmetry, comprising an abundance of interesting connections and ideas, unified by both methods and results’.

The book is divided into nine chapters. The first introduces the concepts of contractability, exchangeability and rotatability and gives representations that are accessible by elementary methods. Martingale methods are used in Chapter 2 to study the paths of exchangeable processes and investigate invariance results for Palm measures. Chapter 3 deals with the theory of convergence in distribution and related approximation properties for exchangeable sequences and processes, including sub-sequence principles. Predictable sampling theorems are presented in Chapter 4 and in Chapter 5 decoupling identities for exchangeable sums and integrals are established. Chapter 6 focusses on exchangeable random sets and the associated excursion theory. Chapters 7 to 9 deal with multivariate symmetries of different kinds. Chapter 7 covers contractable and exchangeable arrays; Chapter 8 deals with rotatable arrays or functionals and the final chapter concentrates on exchangeable random measures on a finite or infinite rectangle.

The 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.

The book is divided into nine chapters. The first introduces the concepts of contractability, exchangeability and rotatability and gives representations that are accessible by elementary methods. Martingale methods are used in Chapter 2 to study the paths of exchangeable processes and investigate invariance results for Palm measures. Chapter 3 deals with the theory of convergence in distribution and related approximation properties for exchangeable sequences and processes, including sub-sequence principles. Predictable sampling theorems are presented in Chapter 4 and in Chapter 5 decoupling identities for exchangeable sums and integrals are established. Chapter 6 focusses on exchangeable random sets and the associated excursion theory. Chapters 7 to 9 deal with multivariate symmetries of different kinds. Chapter 7 covers contractable and exchangeable arrays; Chapter 8 deals with rotatable arrays or functionals and the final chapter concentrates on exchangeable random measures on a finite or infinite rectangle.

The 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.

Reviewer: Neville Weber (Sydney)

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60G09 | Exchangeability for stochastic processes |