Multivariate Characteristic and Correlation Functions (eBook)

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Zoltán Sasvári, Dresden University of Technology, Germany.

Zoltán Sasvári, Technische Universität Dresden.

Preface 5
1 Characteristic functions 11
1.1 Basic properties 11
1.2 Differentiability 19
1.3 Inversion theorems 28
1.4 Basic properties of positive definite functions 35
1.5 Further properties of positive definite functions on Rd 42
1.6 Lévy’s continuity theorem 48
1.7 The theorems of Bochner and Herglotz 51
1.8 Fourier transformation on Rd 57
1.9 Fourier transformation on discrete commutative groups 65
1.10 Basic properties of Gaussian distributions 67
1.11 Some inequalities 71
2 Correlation functions 77
2.1 Random fields 77
2.2 Correlation functions of second order random fields 80
2.3 Continuity and differentiability 85
2.4 Integration with respect to complex measures 87
2.5 The Karhunen-Loéve decomposition 96
2.6 Integration with respect to orthogonal random measures 102
2.7 The theorem of Karhunen 108
2.8 Stationary fields 113
2.9 Spectral representation of stationary fields 119
2.10 Unitary representations 127
2.11 Unitary representations and positive definite functions 135
3 Special properties 142
3.1 Strict positive definiteness 142
3.2 Infinitely differentiable and rapidly decreasing functions 144
3.3 Analytic characteristic functions of one variable 149
3.4 Holomorphic L2 Fourier transforms 158
3.5 Further properties of Gaussian distributions 164
3.6 Fourier transformation of radial measures and functions 170
3.7 Radial characteristic functions 175
3.8 Schoenberg’s theorems on radial characteristic functions 182
3.9 Convex and completely monotone functions 185
3.10 Convolution roots with compact support 194
3.11 Infinitely divisible characteristic functions 197
3.12 Conditionally positive definite functions 199
4 The extension problem 210
4.1 General results 210
4.2 The cases Rd and Zd 218
4.3 Decomposition of locally defined positive definite functions 223
4.4 Extension of radial positive definite functions 231
5 Selected applications 234
5.1 Limit theorems 234
5.2 Sums of independent random vectors and the Jessen-Wintner purity law 236
5.3 Ergodic theorems for stationary fields 244
5.4 Filtration of discrete stationary fields 249
Appendix 252
A Basic notation 252
A.1 Standard notation 252
A.2 Multidimensional notation 253
B Basic analysis 254
B.1 Miscellaneous results from classical analysis 254
B.2 Uniform convergence of continuous functions 266
B.3 Infinite products 271
B.4 Convex functions 274
B.5 The Riemann-Stieltjes integral 276
B.6 Multivariate calculus 277
B.7 The Lebesgue integral on Rd 282
C Advanced analysis 288
C.1 Functions of a complex variable 288
C.2 Almost periodic functions 295
C.3 Fourier series 297
C.4 The Gamma function and the formulae of Stirling and Binet 298
C.5 Bessel functions 306
C.6 The Mellin transform 310
C.7 The Laplace transform 312
C.8 Existence of continuous logarithms 314
C.9 Solutions of certain functional equations 316
C.10 Linear independence of exponential functions 320
D Functional analysis 324
D.1 Inner product spaces 324
D.2 Matrices and kernels 325
D.3 Hilbert spaces and linear operators 337
D.4 Convex sets and the theorem of Krein and Milman 342
D.5 Weak topologies 344
E Measure theory 349
E.1 Borel measures, weak and vague convergence 349
E.2 Convolution of measures and functions 355
F Probability 357
F.1 Basic notions 357
F.2 Convergence of random vectors 360
F.3 Products of probability spaces 362
F.4 Conditional expectation 364
Bibliography 367
Index 371

"[...] There is no book so far devoted entirely to multivariate characteristic functions, which are very important for applications. The present book fills this gap."
Christian Berg, Mathematical Reviews