##
**Group representations in probability and statistics.**
*(English)*
Zbl 0695.60012

Institute of Mathematical Statistics Lecture Notes - Monograph Series 11. Hayward, CA: Institute of Mathematical Statistics (ISBN 0-940600-14-5). vi, 198 p., open access (1988).

The monograph is based on several lectures given by the author during the last years, and the style of lecture notes is preserved in this book: More or less self contained, easily readable even for beginners, numerous examples explaining the main ideas and the possible applications, often examples are discussed before a theorem is stated. The material is not always presented as general as possible, but at a level of generality which is suitable to the whole text.

In most (but not all) examples “groups” are finite groups and therefore the most important mathematical tools are explicit calculations of representations and characters of finite groups, especially of the symmetric groups \(S_ n.\)

A short introduction to representation theory is given in Chapter 2 followed in Chapter 3 by an introduction to random walks including random walks on groups, homogeneous spaces and on Gelfand pairs. The chapter contains interesting examples of (finite) Gelfand pairs and special hints to the literature. A problem which is attacked several times with different tools is the explicit calculation of the rate of convergence of convolution powers to uniform distribution. In Chapter 3 upper and lower bounds are given, in Chapter 4 estimates in terms of strong uniform-times (i.e. special stopping times of the corresponding random walk) are proved.

In the last chapters the author discusses applications to statistics: Various metrics on groups and homogeneous spaces and their statistical use, explicit calculations of the representations of the symmetric group \(S_ n\) and application to spectral analysis (especially time-series, analysis of ranked data and analysis of variance). Chapter 9 is concerned with the fitting of probabilistic models to given data. Here “models” are certain exponential families defined via representations of compact groups.

The reader will find a more or less self contained and unified approach to various problems in probability and statistics connected to (finite or compact) groups. The emphasize is laid on explicit computations or estimations and on concrete applications. As already pointed out the book is full of examples which show the possible applications of group- theoretic methods in probability and statistics. The applications vary from card-shuffling problems, random number generators, partially ranked data in psychology, and coding theory, to tests for uniformity on the sphere \(S^ p\) and the von Mises and Fisher distribution.

In most (but not all) examples “groups” are finite groups and therefore the most important mathematical tools are explicit calculations of representations and characters of finite groups, especially of the symmetric groups \(S_ n.\)

A short introduction to representation theory is given in Chapter 2 followed in Chapter 3 by an introduction to random walks including random walks on groups, homogeneous spaces and on Gelfand pairs. The chapter contains interesting examples of (finite) Gelfand pairs and special hints to the literature. A problem which is attacked several times with different tools is the explicit calculation of the rate of convergence of convolution powers to uniform distribution. In Chapter 3 upper and lower bounds are given, in Chapter 4 estimates in terms of strong uniform-times (i.e. special stopping times of the corresponding random walk) are proved.

In the last chapters the author discusses applications to statistics: Various metrics on groups and homogeneous spaces and their statistical use, explicit calculations of the representations of the symmetric group \(S_ n\) and application to spectral analysis (especially time-series, analysis of ranked data and analysis of variance). Chapter 9 is concerned with the fitting of probabilistic models to given data. Here “models” are certain exponential families defined via representations of compact groups.

The reader will find a more or less self contained and unified approach to various problems in probability and statistics connected to (finite or compact) groups. The emphasize is laid on explicit computations or estimations and on concrete applications. As already pointed out the book is full of examples which show the possible applications of group- theoretic methods in probability and statistics. The applications vary from card-shuffling problems, random number generators, partially ranked data in psychology, and coding theory, to tests for uniformity on the sphere \(S^ p\) and the von Mises and Fisher distribution.

Reviewer: W.Hazod

### MSC:

60B15 | Probability measures on groups or semigroups, Fourier transforms, factorization |

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

60G50 | Sums of independent random variables; random walks |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

62F99 | Parametric inference |