The Normal Distribution

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. Language: English