Stable Probability Measures on Euclidean Spaces and on Locally Compact Groups

Structural Properties and Limit Theorems

  • Wilfried Hazod
  • Eberhard Siebert

Part of the Mathematics and Its Applications book series (MAIA, volume 531)

Table of contents

  1. Front Matter
    Pages N3-xvii
  2. Wilfried Hazod, Eberhard Siebert
    Pages 1-180
  3. Wilfried Hazod, Eberhard Siebert
    Pages 181-425
  4. Wilfried Hazod, Eberhard Siebert
    Pages 427-565
  5. Wilfried Hazod, Eberhard Siebert
    Pages 567-571
  6. Back Matter
    Pages 573-612

About this book

Introduction

Generalising 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. Currently, 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. This will also shed a new light on the classical situation. Chapter 1 provides an introduction to stability problems on vector spaces. Chapter II is concerned with parallel investigations for homogeneous groups and in Chapter III the situation beyond homogeneous Lie groups is treated. Throughout, emphasis is laid on the description of features common to the group- and vector space situation.
Chapter I can be understood by graduate students with some background knowledge in infinite divisibility. Readers of Chapters II and III are assumed to be familiar with basic techniques from probability theory on locally compact groups.

Keywords

Random variable automorphism distribution lie group probability probability theory

Authors and affiliations

  • Wilfried Hazod
    • 1
  • Eberhard Siebert
  1. 1.Mathematical DepartmentUniversity of DortmundDortmundGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-017-3061-7
  • Copyright Information Springer Science+Business Media B.V. 2001
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-481-5832-4
  • Online ISBN 978-94-017-3061-7
  • About this book
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