Overview

Generalizing classical concepts of probability theory, the investigation of operator (semi)-stable laws as possible limit distributions of operator-normalized sums of i.i.d. random variable on finite-dimensional vector space started in 1969. This theory is still in progress and promises interesting applications. Parallel to this, similar stability concepts for probabilities on groups were developed during recent decades. It turns out that the existence of suitable limit distributions has a strong impact on the structure of both the normalizing automorphisms and the underlying group. Indeed, investigations in limit laws led to contractable groups and - at least within the class of connected groups - to homogeneous groups, in particular to groups that are topologically isomorphic to a vector space. Moreover, it has been shown that (semi)-stable measures on groups have a vector space counterpart and vice-versa. The purpose of this book is to describe the structure of limit laws and the limit behaviour of normalized i.i.d. random variables on groups and on finite-dimensional vector spaces from a common point of view.

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9781402000409

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Reviews

...a useful addition to the literature... valuable source of information for the specialists and for those trying to understand some aspects of stability and limit theorems on groups. ... contains an excellent bibliograpy and the authors have included pointers to related developments that are not treated in detail in the text.' Jahresbericht der Deutschen Mathematiker-Vereinigung, 105: 2 (2003)

'...a useful addition to the literature... valuable source of information for the specialists and for those trying to understand some aspects of stability and limit theorems on groups. ... contains an excellent bibliograpy and the authors have included pointers to related developments that are not treated in detail in the text.' Jahresbericht der Deutschen Mathematiker-Vereinigung, 105:2 (2003)

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