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 measures -- 2.5 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 distributions -- 3.6 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 applications -- 5.1 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 -- C.1 Functions of a complex variable -- C.2 Almost periodic functions -- C.3 Fourier series -- C.4 The Gamma function and the formulae of Stirling and Binet -- C.5 Bessel functions -- C.6 The Mellin transform -- C.7 The Laplace transform -- C.8 Existence of continuous logarithms -- C.9 Solutions of certain functional equations -- C.10 Linear independence of exponential functions -- D Functional analysis -- D.1 Inner product spaces -- D.2 Matrices and kernels -- D.3 Hilbert spaces and linear operators -- D.4 Convex sets and the theorem of Krein and Milman -- D.5 Weak topologies -- E Measure theory -- E.1 Borel measures, weak and vague convergence -- E.2 Convolution of measures and functions -- F Probability -- F.1 Basic notions -- F.2 Convergence of random vectors -- F.3 Products of probability spaces -- F.4 Conditional expectation -- Bibliography -- Index.