Multivariate Characteristic and Correlation Functions

Description

In a certain sense characteristic functions and correlation functions are the same, the common underlying concept is positive definiteness. Many results in probability theory, mathematical statistics and stochastic processes can be derived by using these functions. While there are books on characteristic functions of one variable, books devoting some sections to the multivariate case, and books treating the general case of locally compact groups, interestingly there is no book devoted entirely to the multidimensional case which is extremely important for applications. This book is intended to fill this gap at least partially. It makes the basic concepts and results on multivariate characteristic and correlation functions easily accessible to both students and researchers in a comprehensive manner. The first chapter presents basic results and should be read carefully since it is essential for the understanding of the subsequent chapters. The second chapter is devoted to correlation functions, their applications to stationary processes and some connections to harmonic analysis. In Chapter 3 we deal with several special properties, Chapter 4 is devoted to the extension problem while Chapter 5 contains a few applications. A relatively large appendix comprises topics like infinite products, functional equations, special functions or compact operators. 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