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The Normal Distribution

Characterizations with Applications

  • Wlodzimierz Bryc

Part of the Lecture Notes in Statistics book series (LNS, volume 100)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Wlodzimierz Bryc
    Pages 1-3
  3. Wlodzimierz Bryc
    Pages 5-21
  4. Wlodzimierz Bryc
    Pages 23-38
  5. Wlodzimierz Bryc
    Pages 39-49
  6. Wlodzimierz Bryc
    Pages 51-69
  7. Wlodzimierz Bryc
    Pages 71-80
  8. Wlodzimierz Bryc
    Pages 81-91
  9. Wlodzimierz Bryc
    Pages 93-107
  10. Wlodzimierz Bryc
    Pages 109-121
  11. Back Matter
    Pages 123-142

About this book

Introduction

This book is a concise presentation of the normal distribution on the real line and its counterparts on more abstract spaces, which we shall call the Gaussian distributions. The material is selected towards presenting characteristic properties, or characterizations, of the normal distribution. There are many such properties and there are numerous rel­ evant works in the literature. In this book special attention is given to characterizations generated by the so called Maxwell's Theorem of statistical mechanics, which is stated in the introduction as Theorem 0.0.1. These characterizations are of interest both intrin­ sically, and as techniques that are worth being aware of. The book may also serve as a good introduction to diverse analytic methods of probability theory. We use characteristic functions, tail estimates, and occasionally dive into complex analysis. In the book we also show how the characteristic properties can be used to prove important results about the Gaussian processes and the abstract Gaussian vectors. For instance, in Section 5.4 we present Fernique's beautiful proofs of the zero-one law and of the integrability of abstract Gaussian vectors. The central limit theorem is obtained via characterizations in Section 7.3.

Keywords

Gaussian distribution Gaussian process Moment Random variable Rang Variance mixing normal distribution probability probability theory statistics uniform integrability

Authors and affiliations

  • Wlodzimierz Bryc
    • 1
  1. 1.Department of MathematicsUniversity of CincinnatiCincinnatiUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-2560-7
  • Copyright Information Springer-Verlag New York 1995
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-387-97990-8
  • Online ISBN 978-1-4612-2560-7
  • Series Print ISSN 0930-0325
  • Buy this book on publisher's site