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The Normal Distribution Characterizations with Applications

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...

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Bibliographic Details
Main Author: Bryc, Wlodzimierz
Format: eBook
Language:English
Published: New York, NY Springer New York 1995, 1995
Edition:1st ed. 1995
Series:Lecture Notes in Statistics
Subjects:
Probability Theory And Stochastic Processes
Probabilities
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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
  • 2.7 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
  • 4.3 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
  • 6.2 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
  • 7.4 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

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