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Limit Theorems of Probability Theory

  • Book
  • © 2000

Overview

Editors:
  1. Yu. V. Prokhorov

    1. Steklov Mathematical Institute, Moscow, Russia

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  2. V. Statulevičius

    1. Institute of Mathematics and Informatics, Lithuanian Academy of Sciences, Vilnius, Lithuania

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    You can also search for this editor in PubMed   Google Scholar

  • The book covers aspects of limit theorems by a well-known group of researchers.

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Table of contents (5 chapters)

  1. Front Matter

    Pages I-X
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  2. Classical-Type Limit Theorems for Sums of Independent Random Variables

    • V. V. Petrov
    Pages 1-24

  3. The Accuracy of Gaussian Approximation in Banach Spaces

    • V. Bentkus, F. Götze, V. Paulauskas, A. Račkauskas
    Pages 25-111

  4. Approximation of Distributions of Sums of Weakly Dependent Random Variables by the Normal Distribution

    • J. Sunklodas
    Pages 113-165

  5. Refinements of the Central Limit Theorem for Homogeneous Markov Chains

    • P. Gudynas
    Pages 167-183

  6. Limit Theorems on Large Deviations

    • L. Saulis, V. Statulevičius
    Pages 185-266

  7. Back Matter

    Pages 267-275
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Keywords

About this book

This book consists of five parts written by different authors devoted to various problems dealing with probability limit theorems. The first part, "Classical-Type Limit Theorems for Sums ofIndependent Random Variables" (V.v. Petrov), presents a number of classical limit theorems for sums of independent random variables as well as newer related results. The presentation dwells on three basic topics: the central limit theorem, laws of large numbers and the law of the iterated logarithm for sequences of real-valued random variables. The second part, "The Accuracy of Gaussian Approximation in Banach Spaces" (V. Bentkus, F. G6tze, V. Paulauskas and A. Rackauskas), reviews various results and methods used to estimate the convergence rate in the central limit theorem and to construct asymptotic expansions in infinite-dimensional spaces. The authors con­ fine themselves to independent and identically distributed random variables. They do not strive to be exhaustive or to obtain the most general results; their aim is merely to point out the differences from the finite-dimensional case and to explain certain new phenomena related to the more complex structure of Banach spaces. Also reflected here is the growing tendency in recent years to apply results obtained for Banach spaces to asymptotic problems of statistics.

Reviews

"The book is a very fine collection of five authoritative essays on five main directions in asymptotic probability, each giving a state-of-the-art overview as seen from Vilnius ... in 1991, when the original Russian edition was published in Moscow. Of course, any insider appreciates the relevance, indeed, the importance of the Vilnius view. The nine-year delay with this English translation (a superb job, done by B.Seckler) is unessential: while all five fields continued to develop and grow in these nine years, the essayists make honest attempts to explore the very essence of their fields, and mostly with success. Thus, most later developments of note would easily fit into the framework described in each of the five directions. ... People doing research in any of the areas above will find the volume extremely useful as a reference work that, in particular, makes hundreds of papers from Russian periodicals available for the first time. It will be practically indispensable for a graduate student beginning to do research in asymptotic probability."

Sándor Csörgö, Acta Scientiarum Mathematicarum 2002, Vol. 68, pp. 1013-1014

"... This book should be at hand for any specialist on asymptotic methods in probability and statistics, and can also serve as a basic reference for graduate and Ph.D. students of stochastics."

European Mathematical Society, March 2002

"The monograph is written at the research level and will be a useful reference for graduate students and teachers."

Journal of the American Statistical Association

Editors and Affiliations

  • Steklov Mathematical Institute, Moscow, Russia

    Yu. V. Prokhorov

  • Institute of Mathematics and Informatics, Lithuanian Academy of Sciences, Vilnius, Lithuania

    V. Statulevičius

Bibliographic Information

  • Book Title: Limit Theorems of Probability Theory

  • Editors: Yu. V. Prokhorov, V. Statulevičius

  • DOI: https://doi.org/10.1007/978-3-662-04172-7

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2000

  • Hardcover ISBN: 978-3-540-57045-5Published: 28 August 2000

  • Softcover ISBN: 978-3-642-08170-5Published: 01 December 2010

  • eBook ISBN: 978-3-662-04172-7Published: 14 March 2013

  • Edition Number: 1

  • Number of Pages: X, 273

  • Additional Information: Original Russian edition published by VINITI, Moscow, 1991

  • Topics: Probability Theory and Stochastic Processes

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