Asymptotic Behaviour of Linearly Transformed Sums of Random Variables

  • Valery Buldygin
  • Serguei Solntsev

Part of the Mathematics and Its Applications book series (MAIA, volume 416)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Random series and linear transformations of sequences of independent random elements

    1. Front Matter
      Pages xv-xv
    2. Valery Buldygin, Serguei Solntsev
      Pages 1-46
    3. Valery Buldygin, Serguei Solntsev
      Pages 47-121
  3. Limit theorems for operator-normed sums of independent random vectors and their applications

    1. Front Matter
      Pages 215-215
    2. Valery Buldygin, Serguei Solntsev
      Pages 217-268
    3. Valery Buldygin, Serguei Solntsev
      Pages 269-306
    4. Valery Buldygin, Serguei Solntsev
      Pages 307-341
    5. Valery Buldygin, Serguei Solntsev
      Pages 343-362
    6. Valery Buldygin, Serguei Solntsev
      Pages 363-416
  4. Back Matter
    Pages 443-504

About this book

Introduction

Limit theorems for random sequences may conventionally be divided into two large parts, one of them dealing with convergence of distributions (weak limit theorems) and the other, with almost sure convergence, that is to say, with asymptotic prop­ erties of almost all sample paths of the sequences involved (strong limit theorems). Although either of these directions is closely related to another one, each of them has its own range of specific problems, as well as the own methodology for solving the underlying problems. This book is devoted to the second of the above mentioned lines, which means that we study asymptotic behaviour of almost all sample paths of linearly transformed sums of independent random variables, vectors, and elements taking values in topological vector spaces. In the classical works of P.Levy, A.Ya.Khintchine, A.N.Kolmogorov, P.Hartman, A.Wintner, W.Feller, Yu.V.Prokhorov, and M.Loeve, the theory of almost sure asymptotic behaviour of increasing scalar-normed sums of independent random vari­ ables was constructed. This theory not only provides conditions of the almost sure convergence of series of independent random variables, but also studies different ver­ sions of the strong law of large numbers and the law of the iterated logarithm. One should point out that, even in this traditional framework, there are still problems which remain open, while many definitive results have been obtained quite recently.

Keywords

Gaussian process Markov process Probability theory Random variable mathematical statistics statistics

Authors and affiliations

  • Valery Buldygin
    • 1
  • Serguei Solntsev
    • 1
  1. 1.Department of Higher MathematicsKiev Polytechnic InstituteKievUkraine

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-011-5568-7
  • Copyright Information Kluwer Academic Publishers 1997
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-94-010-6346-3
  • Online ISBN 978-94-011-5568-7
  • About this book
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