Author | Bryc, Wlodzimierz. author |
---|---|
Title | The Normal Distribution [electronic resource] : Characterizations with Applications / by Wlodzimierz Bryc |
Imprint | New York, NY : Springer New York, 1995 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-2560-7 |
Descript | VIII, 139 p. online resource |
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 Bernsteinโs 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 Levyโs 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