Stable Convergence and Stable Limit Theorems

1. Front Matter

   Pages i-x

   PDF

2. Why Stable Convergence?

   • Erich Häusler, Harald Luschgy

   Pages 1-9

3. Weak Convergence of Markov Kernels

   • Erich Häusler, Harald Luschgy

   Pages 11-19

4. Stable Convergence of Random Variables

   • Erich Häusler, Harald Luschgy

   Pages 21-37

5. Applications

   • Erich Häusler, Harald Luschgy

   Pages 39-53

6. Stability of Limit Theorems

   • Erich Häusler, Harald Luschgy

   Pages 55-65

7. Stable Martingale Central Limit Theorems

   • Erich Häusler, Harald Luschgy

   Pages 67-122

8. Stable Functional Martingale Central Limit Theorems

   • Erich Häusler, Harald Luschgy

   Pages 123-144

9. A Stable Limit Theorem with Exponential Rate

   • Erich Häusler, Harald Luschgy

   Pages 145-158

10. Autoregression of Order One

    • Erich Häusler, Harald Luschgy

    Pages 159-172

11. Galton-Watson Branching Processes

    • Erich Häusler, Harald Luschgy

    Pages 173-186

12. Back Matter

    Pages 187-228

    PDF

The authors present a concise but complete exposition of the mathematical theory of stable convergence and give various applications in different areas of probability theory and mathematical statistics to illustrate the usefulness of this concept. Stable convergence holds in many limit theorems of probability theory and statistics – such as the classical central limit theorem – which are usually formulated in terms of convergence in distribution. Originated by Alfred Rényi, the notion of stable convergence is stronger than the classical weak convergence of probability measures. A variety of methods is described which can be used to establish this stronger stable convergence in many limit theorems which were originally formulated only in terms of weak convergence. Naturally, these stronger limit theorems have new and stronger consequences which should not be missed by neglecting the notion of stable convergence. The presentation will be accessible to researchers and advanced students at the master's level with a solid knowledge of measure theoretic probability.

• 60-02, 60F05, 60F17
• Gauss kernels
• limit theorems
• mixing convergence of random variables
• stable convergence of random variables
• weak convergence of Markov kernels

“This book presents an account of stable convergence and stable limit theorems which can serve as an introduction to the area. … The book is a big account of all major stable limit theorems which have been established in the last 50 years or so.” (Nikolai N. Leonenko, zbMATH 1356.60004, 2017)

“The present book represents a comprehensive account of the theory of stable convergence. The theory is illustrated by a number of examples and applied to a variety of limit theorems. … The book is well written, and the concepts are clearly explained. I enjoyed reading it because of both the contents and the authors’ attractive style of presentation. … I concur with this and think that the book will appeal to the student as much as to the specialist.” (Alexander Iksanov, Mathematical Reviews, February, 2016)

• Mathematical Institute, University of Giessen, Giessen, Germany

  Erich Häusler

• University of Trier, Trier, Germany

  Harald Luschgy

Erich Haeusler studied mathematics and physics at the University of Bochum from 1972 to 1978. He received his doctorate in mathematics in 1982 from the University of Munich. Since 1991 he has been Professor of Mathematics at the University of Giessen, where he teaches probability and mathematical statistics. Harald Luschgy studied mathematics, physics and mathematical logic at the Universities of Bonn and Münster. He received his doctorate in mathematics in 1976 from the University of Münster. He held visiting positions at the Universities of Hamburg, Bayreuth, Dortmund, Oldenburg, Passau and Wien and was a recipient of a Heisenberg grant from the DFG. Since 1995 he is Professor of Mathematics at the University of Trier where he teaches probability and mathematical statistics.

Policies and ethics