Fourier Analysis in Probability Theory (Probability and mathematical statistics)

14.4 Convergence of Random Power Series with Identically and Independently......Page 0
1.1 Measurable Space; Probability Space......Page 14
1.2 Measurable Functions; Random Variables......Page 19
1.3 Product Space......Page 20
1.4 Integrals......Page 21
1.5 The Fubini-Tonelli Theorem......Page 25
1.6 Integrals on -R,......Page 26
1.7 Functions of Bounded Variation......Page 29
1.8 Signed Measure; Decomposition Theorems......Page 31
1.9 The Lebesgue Integral on Rt......Page 33
1.10 Inequalities......Page 37
1.11 Convex Functions......Page 40
1.12 Analytic Functions......Page 41
1.13 Jensen's and Carleman's Theorems......Page 44
1.14 Analytic Continuation......Page 45
1.15 Maximum Modulus Theorem and Theorems of Phragmen-Lindelof . .......Page 48
1.16 Inner Product Space......Page 50
2.1 The Riemann-Lebesgue Lemma......Page 56
2.2 Fourier Series......Page 62
2.3 The Fourier Transform of a Function in L'( —oo, oo)......Page 65
2.4 Magnitude of Fourier Coefficients; the Continuity Modulus......Page 66
2.5 More about the Magnitude of Fourier Coefficients......Page 69
2.6 Some Elementary Lemmas......Page 72
2.7 Continuity and Magnitude of Fourier Transforms......Page 75
2.8 Operations on Fourier Series......Page 77
2.9 Operations on Fourier Transforms......Page 80
2.10 Completeness of Trigonometric Functions......Page 82
2.11 Unicity Theorem for Fourier Transforms......Page 86
2.12 Fourier Series and Fourier Transform of Convolutions......Page 87
Notes......Page 91
3.1 Monotone Functions and Distribution Functions......Page 94
3.2 Fourier-Stieltjes Series......Page 97
3.3 Average of Fourier-Stieltjes Coefficients......Page 99
3.4 Unicity Theorem for Fourier-Stieltjes Coefficients......Page 101
3.5 Fourier-Stieltjes Transform and Characteristic Function......Page 102
3.6 Periodic Characteristic Functions......Page 106
3.7 Some Inequality Relations for Characteristic Functions......Page 108
3.8 Average of a Characteristic Function......Page 115
3.9 Convolution of Nondecreasing Functions......Page 117
Convolution......Page 121
4.1 Convergence of Fourier Series......Page 126
4.2 Convergence of Fourier-Stieltjes Series......Page 134
4.3 Fourier's Integral Theorems; Inversion Formulas for Fourier Transforms......Page 136
4.4 Inversion Formula for Fourier-Stieltjes Transforms......Page 141
4.5 Summability......Page 144
4.6 (C,l)-Summability for Fourier Series......Page 148
4.8 Summability Theorems for Fourier Transforms......Page 158
ing Function......Page 164
Transform......Page 167
4.11 Some Examples, Using Fourier Transforms......Page 171
Notes......Page 177
5.1 Nature of the Problems......Page 179
5.2 Some General Convergence Theorems I......Page 180
5.3 Some General Convergence Theorems II......Page 187
5.4 General Convergence Theorems for the Stieltjes Integral......Page 191
5.5 Wiener's Formula......Page 194
Distribution Function......Page 199
Notes......Page 205
6.1 Fourier Series in an Inner Product Space......Page 207
( —oo, oo)......Page 214
of Analytic Functions......Page 223
6.4 A Theorem of Szego and Smirnov......Page 227
of Analytic Functions......Page 231
6.6 A Theorem of Paley and Wiener......Page 241
Notes......Page 244
7.1 The Laplace Transform......Page 245
7.2 The Convergence Abscissa......Page 251
7.3 Analyticity of a Laplace-Stieltjes Transform......Page 255
7.4 Inversion Formulas for Laplace Transforms......Page 259
7.5 The Laplace Transform of a Convolution......Page 265
7.6 Operations of Laplace Transforms and Some Examples......Page 272
7.7 The Bilateral Laplace-Stieltjes Transform......Page 278
7.8 Mellin-Stieltjes Transforms......Page 282
7.9 The Mellin Transform......Page 286
8.1 A Theorem of Hardy......Page 291
8.2 A Theorem of Paley and Wiener on Exponential Entire Functions ... .......Page 297
8.3 Theorems of Ingham and Levinson......Page 301
8.4 Singularities of Laplace Transforms......Page 311
8.5 Abelian Theorems for Laplace Transforms......Page 314
8.6 Tauberian Theorems......Page 317
8.7 Multiple Fourier Series and Transforms......Page 327
8.8 Nondecreasing Functions and Distribution Functions in Rm......Page 336
8.9 The Multiple Fourier-Stieltjes Transform......Page 339
Notes......Page 341
9.1 Helly Theorems and Convergence of Nondecreasing Functions......Page 343
9.2 Convergence of Distribution Functions with Bounded Spectra......Page 352
9.3 Convergence of Distribution Functions......Page 354
of a Characteristic Function......Page 366
9.5 A Basic Theorem on Analytic Characteristic Functions......Page 369
9.6 Continuity Theorems on Intervals and Uniqueness Theorems......Page 370
9.7 The Compact Set of Characteristic Functions......Page 375
Notes......Page 378
10.1 Characteristic Properties of Fourier Coefficients......Page 379
10.2 Basic Theorems on Characterization of a Characteristic Function ... .......Page 385
103. Characteristic Properties of Characteristic Functions......Page 389
10.4 Functions of the Wiener Class......Page 396
10.5 Some Sufficient Criteria for Characteristic Functions......Page 398
10.6 More Criteria for Characteristic Functions......Page 405
Notes......Page 410
11.1 Moments, Basic Properties......Page 413
11.2 Smoothness of a Characteristic Function and the Existence of Moments......Page 421
11.4 Absolute Moments......Page 441
11.5 Boundedness of the Spectra of Distribution Functions......Page 444
11.6 Integrable Characteristic Functions......Page 450
11.7 Analyticity of Distribution Functions......Page 452
11.8 Mean Concentration Function of a Distribution Function......Page 458
11.9 Some Properties of Analytic Characteristic Functions......Page 465
11.10 Characteristic Functions Analytic in the Half-Plane......Page 469
11.11 Entire Characteristic Functions I......Page 473
11.12 Entire Characteristic Functions II......Page 479
Notes......Page 486
12.1 Convergence of a Sequence of Random Variables......Page 488
12.2 The Borel Theorem......Page 499
12.3 The Zero-One Law......Page 501
12.4 The Equivalence Theorem......Page 505
12.5 The Three Series Theorem......Page 509
12.6 Sufficient Conditions for the Convergence of a Series......Page 514
12.7 Convergence Criteria and the Typical Function......Page 517
12.8 Rademacher and Steinhaus Functions......Page 520
12.9 Convergence of Products of Characteristic Functions......Page 526
12.10 Unconditional Convergence......Page 532
12.11 Absolute Convergence......Page 539
12.12 Essential Convergence......Page 543
Notes......Page 547
13.1 Continuity and Discontinuity Properties of the Sum of a Series ... "......Page 549
13.2 Integrability of the Sum of a Series......Page 556
13.3 Magnitude of the Characteristic Functions of the Sums of Series . . .......Page 559
Functions of Singular Distributions......Page 569
13.5 Further Theorems on Rademacher Series......Page 578
13.6 Sums of Independent Random Variables......Page 586
13.7 Convergent Systems......Page 596
13.8 Integrability of Sums of Series; Strong and Weak Convergences of Series......Page 599
13.9 Vanishing of the Sum of a Series......Page 603
13.10 Summability of Series......Page 609
14.1 Fourier Series with Rademacher Coefficients......Page 618
14.2 Random Fourier Series......Page 625
14.3 Random Power Series, Convergence......Page 632
Distributed Random Coefficients......Page 638
14.5 Analytic Continuation of Random Power Series......Page 641
14.6 Fourier Series with Orthogonal Random Coefficients......Page 647
Notes......Page 653
References......Page 655
INDEX......Page 674