# The Normal Distribution

## Characterizations with Applications

• Wlodzimierz Bryc
Book

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

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

### 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
• 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