Stable Convergence and Stable Limit Theorems [Hardcover]

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 R?nyi, 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.Preface.- 1.Weak Convergence of Markov Kernels.- 2.Stable Convergence.- 3.Applications.- 4.Stability of Limit Theorems.- 5.Stable Martingale Central Limit Theorems.- 6.Stable Functional Martingale Central Limit Theorems.- 7.A Stable Limit Theorem with Exponential Rate.- 8.Autoregression of Order One.- 9.Branching Processes.- A. Appendix.- B. Appendix.- Bibliography.

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 tlw