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**Stable probability measures on Euclidean spaces and locally compact groups. Structural properties and limit theorems.**
*(English)*
Zbl 1002.60002

Mathematics and its Applications (Dordrecht). 531. Dordrecht: Kluwer Academic Publishers. xvii, 612 p. (2001).

In this comprehensive monograph, which is about the only one which treats the topic in such generality, the authors offer a detailed presentation on structural properties of stable probability measures on Euclidean spaces and in the much more general setting of locally compact groups. The book is well written and serves as an introduction to the above mentioned topic for researchers and graduate students and – presenting a large part of the investigations of the last thirty years from a common point of view – also as a reference text for specialists. During the early stages of the preparation of the book, the second named author Eberhard Siebert suddenly died, but nevertheless Wilfried Hazod decided to continue the joint work.

The emphasis of the book is to present structural properties of stable laws on various algebraic structures from a common point of view, rather than central limit theorems. It has three main chapters, each of which is divided into several sections. It follows to some extend the historical development of the topic. In Chapter I stability on finite-dimensional real vector spaces is considered, including operator stable and operator semistable laws. Using an approach similar to the vector space case, the concept of stability on simply connected nilpotent Lie groups is developed in Chapter II, the largest part of the monograph. Finally, in Chapter III, the most general case of stable laws on general locally compact groups is treated. Each chapter is rounded up by a concluding “References and comments” section outlining the sources of results presented (so giving a useful historical survey of the relevant literature), and suggesting open problems awaiting further investigation. There are the impressive number of 452 references listed.

The emphasis of the book is to present structural properties of stable laws on various algebraic structures from a common point of view, rather than central limit theorems. It has three main chapters, each of which is divided into several sections. It follows to some extend the historical development of the topic. In Chapter I stability on finite-dimensional real vector spaces is considered, including operator stable and operator semistable laws. Using an approach similar to the vector space case, the concept of stability on simply connected nilpotent Lie groups is developed in Chapter II, the largest part of the monograph. Finally, in Chapter III, the most general case of stable laws on general locally compact groups is treated. Each chapter is rounded up by a concluding “References and comments” section outlining the sources of results presented (so giving a useful historical survey of the relevant literature), and suggesting open problems awaiting further investigation. There are the impressive number of 452 references listed.

Reviewer: H.-P.Scheffler (Dortmund)

### MSC:

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

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

60E07 | Infinitely divisible distributions; stable distributions |

60F05 | Central limit and other weak theorems |