Multivariate Characteristic and Correlation Functions - Sasvári, Zoltán (De Gruyter, 2013)
E.1 Borel measures, weak and vague convergence
Preface; 1 Characteristic functions; 1.1 Basic properties; 1.2 Differentiability; 1.3 Inversion theorems; 1.4 Basic properties of positive definite functions; 1.5 Further properties of positive definite functions on Rd; 1.6 Lévy's continuity theorem; 1.7 The theorems of Bochner and Herglotz; 1.8 Fourier transformation on Rd; 1.9 Fourier transformation on discrete commutative groups; 1.10 Basic properties of Gaussian distributions; 1.11 Some inequalities; 2 Correlation functions; 2.1 Random fields; 2.2 Correlation functions of second order random fields; 2.3 Continuity and differentiability
2.4 Integration with respect to complex measures2.5 The Karhunen-Loéve decomposition; 2.6 Integration with respect to orthogonal random measures; 2.7 The theorem of Karhunen; 2.8 Stationary fields; 2.9 Spectral representation of stationary fields; 2.10 Unitary representations; 2.11 Unitary representations and positive definite functions; 3 Special properties; 3.1 Strict positive definiteness; 3.2 Infinitely differentiable and rapidly decreasing functions; 3.3 Analytic characteristic functions of one variable; 3.4 Holomorphic L2 Fourier transforms
3.5 Further properties of Gaussian distributions3.6 Fourier transformation of radial measures and functions; 3.7 Radial characteristic functions; 3.8 Schoenberg's theorems on radial characteristic functions; 3.9 Convex and completely monotone functions; 3.10 Convolution roots with compact support; 3.11 Infinitely divisible characteristic functions; 3.12 Conditionally positive definite functions; 4 The extension problem; 4.1 General results; 4.2 The cases Rd and Zd; 4.3 Decomposition of locally defined positive definite functions; 4.4 Extension of radial positive definite functions
5 Selected applications5.1 Limit theorems; 5.2 Sums of independent random vectors and the Jessen-Wintner purity law; 5.3 Ergodic theorems for stationary fields; 5.4 Filtration of discrete stationary fields; Appendix; A Basic notation; A.1 Standard notation; A.2 Multidimensional notation; B Basic analysis; B.1 Miscellaneous results from classical analysis; B.2 Uniform convergence of continuous functions; B.3 Infinite products; B.4 Convex functions; B.5 The Riemann-Stieltjes integral; B.6 Multivariate calculus; B.7 The Lebesgue integral on Rd; C Advanced analysis