"The book sets recent work in context and represents a valuable addition to the standard texts on characterizations." - J.R. Leslie, Macquarie University, Sydney, Australia

• 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.

"The book sets recent work in context and represents a valuable addition to the standard texts on characterizations." - J.R. Leslie, Macquarie University, Sydney, Australia

• 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.

"The book sets recent work in context and represents a valuable addition to the standard texts on characterizations." - J.R. Leslie, Macquarie University, Sydney, Australia

• 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.