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.

Inhalt

1 Probability tools.- 1.1 Moments.- 1.2 Lp-spaces.- 1.3 Tail estimates.- 1.4 Conditional expectations.- 1.5 Characteristic functions.- 1.6 Symmetrization.- 1.7 Uniform integrability.- 1.8 The Mellin transform.- 1.9 Problems.- 2 Normal distributions.- 2.1 Univariate normal distributions.- 2.2 Multivariate normal distributions.- 2.3 Analytic characteristic functions.- 2.4 Hermite expansions.- 2.5 Cramer and Marcinkiewicz theorems.- 2.6 Large deviations.- 2.6.1 A numerical example.- 2.7 Problems.- 3 Equidistributed linear forms.- 3.1 Two-stability.- 3.2 Measures on linear spaces.- 3.3 Linear forms.- 3.4 Exponential analogy.- 3.5 Exponential distributions on lattices.- 3.6 Problems.- 4 Rotation invariant distributions.- 4.1 Spherically symmetric vectors.- 4.2 Rotation invariant absolute moments.- 4.2.1 Proof of Theorem 4.2.2 for p = 1.- 4.2.2 Proof of Theorem 4.2.2 in the general case.- 4.2.3 Pairs of random variables.- 4.3 Infinite spherically symmetric sequences.- 4.4 Problems.- 5 Independent linear forms.- 5.1 Bernstein's theorem.- 5.2 Gaussian distributions on groups.- 5.3 Independence of linear forms.- 5.4 Strongly Gaussian vectors.- 5.5 Joint distributions.- 5.6 Problems.- 6 Stability and weak stability.- 6.1 Coefficients of dependence.- 6.1.1 Normal case.- 6.2 Weak stability.- 6.3 Stability.- 6.4 Problems.- 7 Conditional moments.- 7.1 Finite sequences.- 7.2 Extension of Theorem 7.1.2.- 7.3 Central Limit Theorem.- 7.3.1 CLT for i. i. d. sums.- 7.4 Empirical mean and variance.- 7.5 Infinite sequences and conditional moments.- 7.6 Problems.- 8 Gaussian processes.- 8.1 Construction of the Wiener process.- 8.2 Levy's characterization theorem.- 8.3 Arbitrary trajectories.- 8.4 Second order conditional structure.- A Solutions of selected problems.- A.1 Solutions for Chapter 1.- A.2 Solutions for Chapter 2.- A.3 Solutions for Chapter 3.- A.4 Solutions for Chapter 4.- A.5 Solutions for Chapter 5.- A.6 Solutions for Chapter 6.- A.7 Solutions for Chapter 7.