Fourier Analysis in Probability Theory (Probability and mathematical statistics)

Contents......Page f005.djvu
Preface......Page f011.djvu
1.1 Measurable Space; Probability Space......Page p001.djvu
1.2 Measurable Functions; Random Variables......Page p006.djvu
1.3 Product Space......Page p007.djvu
1.4 Integrals......Page p008.djvu
1.5 The Fubini-Tonelli Theorem......Page p012.djvu
1.6 Integrals on Rt......Page p013.djvu
1.7 Functions of Bounded Variation......Page p016.djvu
1.8 Signed Measure; Decomposition Theorems......Page p018.djvu
1.9 The Lebesgue Integral on Rx......Page p020.djvu
1.10 Inequalities......Page p024.djvu
1.11 Convex Functions......Page p027.djvu
1.12 Analytic Functions......Page p028.djvu
1.13 Jensen\'s and Carleman\'s Theorems......Page p031.djvu
1.14 Analytic Continuation......Page p032.djvu
1.15 Maximum Modulus Theorem and Theorems of Phragmen-Lindelof......Page p035.djvu
1.16 Inner Product Space......Page p037.djvu
2.1 The Riemann-Lebesgue Lemma......Page p043.djvu
2.2 Fourier Series......Page p049.djvu
2.3 The Fourier Transform of a Function in L^1(-Infty, Infty)......Page p052.djvu
2.4 Magnitude of Fourier Coefficients; the Continuity Modulus......Page p053.djvu
2.5 More about the Magnitude of Fourier Coefficients......Page p056.djvu
2.6 Some Elementary Lemmas......Page p059.djvu
2.7 Continuity and Magnitude of Fourier Transforms......Page p062.djvu
2.8 Operations on Fourier Series......Page p064.djvu
2.9 Operations on Fourier Transforms......Page p067.djvu
2.10 Completeness of Trigonometric Functions......Page p069.djvu
2.11 Unicity Theorem for Fourier Transforms......Page p073.djvu
2.12 Fourier Series and Fourier Transform of Convolutions......Page p074.djvu
Notes......Page p078.djvu
3.1 Monotone Functions and Distribution Functions......Page p081.djvu
3.2 Fourier-Stieltjes Series......Page p084.djvu
3.3 Average of Fourier-Stieltjes Coefficients......Page p086.djvu
3.4 Unicity Theorem for Fourier-Stieltjes Coefficients......Page p088.djvu
3.5 Fourier-Stieltjes Transform and Characteristic Function......Page p089.djvu
3.6 Periodic Characteristic Functions......Page p093.djvu
3.7 Some Inequality Relations for Characteristic Functions......Page p095.djvu
3.8 Average of a Characteristic Function......Page p102.djvu
3.9 Convolution of Nondecreasing Functions......Page p104.djvu
3.10 The Fourier-Stieltjes Transform of a Convolution and the Bernoulli Convolution......Page p108.djvu
Notes......Page p111.djvu
4.1 Convergence of Fourier Series......Page p113.djvu
4.2 Convergence of Fourier-Stieltjes Series......Page p121.djvu
4.3 Fourier\'s Integral Theorems; Inversion Formulas for Fourier Transforms......Page p123.djvu
4.4 Inversion Formula for Fourier-Stieltjes Transforms......Page p128.djvu
4.5 Summability......Page p131.djvu
4.6 (C, 1)-Summability for Fourier Series......Page p135.djvu
4.7 Abel-Summability for Fourier Series......Page p140.djvu
4.8 Summability Theorems for Fourier Transforms......Page p145.djvu
4.9 Determination of the Absolutely Continuous Component of a Nondecreasing Function......Page p151.djvu
4.10 Fourier Series and Approximate Fourier Series of a Fourier-Stieltjes Transform......Page p154.djvu
4.11 Some Examples, Using Fourier Transforms......Page p158.djvu
Notes......Page p164.djvu
5.1 Nature of the Problems......Page p166.djvu
5.2 Some General Convergence Theorems I......Page p167.djvu
5.3 Some General Convergence Theorems II......Page p174.djvu
5.4 General Convergence Theorems for the Stieltjes Integral......Page p178.djvu
5.5 Wiener\'s Formula......Page p181.djvu
5.6 Applications of General Convergence Theorems to the Estimates of a Distribution Function......Page p186.djvu
Notes......Page p192.djvu
6.1 Fourier Series in an Inner Product Space......Page p194.djvu
6.2 Fourier Transform of a Function in L^2(-Infty, Infty)......Page p201.djvu
6.3 The Class H^2 of Analytic Functions......Page p210.djvu
6.4 A Theorem of Szego and Smirnov......Page p214.djvu
6.5 The Class mathfrak{H}^2 of Analytic Functions......Page p218.djvu
6.6 A Theorem of Paley and Wiener......Page p228.djvu
7.1 The Laplace Transform......Page p232.djvu
7.2 The Convergence Abscissa......Page p238.djvu
7.3 Analyticity of a Laplace-Stieltjes Transform......Page p242.djvu
7.4 Inversion Formulas for Laplace Transforms......Page p246.djvu
7.5 The Laplace Transform of a Convolution......Page p252.djvu
7.6 Operations of Laplace Transforms and Some Examples......Page p259.djvu
7.7 The Bilateral Laplace-Stieltjes Transform......Page p265.djvu
7.8 Mellin-Stieltjes Transforms......Page p269.djvu
7.9 The Mellin Transform......Page p273.djvu
Notes......Page p276.djvu
8.1 A Theorem of Hardy......Page p278.djvu
8.2 A Theorem of Paley and Wiener on Exponential Entire Functions......Page p284.djvu
8.3 Theorems of Ingham and Levinson......Page p288.djvu
8.4 Singularities of Laplace Transforms......Page p298.djvu
8.5 Abelian Theorems for Laplace Transforms......Page p301.djvu
8.6 Tauberian Theorems......Page p304.djvu
8.7 Multiple Fourier Series and Transforms......Page p314.djvu
8.8 Nondecreasing Functions and Distribution Functions in Rm......Page p323.djvu
8.9 The Multiple Fourier-Stieltjes Transform......Page p326.djvu
Notes......Page p328.djvu
9.1 Helly Theorems and Convergence of Nondecreasing Functions......Page p330.djvu
9.2 Convergence of Distribution Functions with Bounded Spectra......Page p339.djvu
9.3 Convergence of Distribution Functions......Page p341.djvu
9.4 Continuous Distribution Functions: A General Integral Transform of a Characteristic Function......Page p353.djvu
9.5 A Basic Theorem on Analytic Characteristic Functions......Page p356.djvu
9.6 Continuity Theorems on Intervals and Uniqueness Theorems......Page p357.djvu
9.7 The Compact Set of Characteristic Functions......Page p362.djvu
Notes......Page p365.djvu
10.1 Characteristic Properties of Fourier Coefficients......Page p366.djvu
10.2 Basic Theorems on Characterization of a Characteristic Function......Page p372.djvu
10.3 Characteristic Properties of Characteristic Functions......Page p376.djvu
10.4 Functions of the Wiener Class......Page p383.djvu
10.5 Some Sufficient Criteria for Characteristic Functions......Page p385.djvu
10.6 More Criteria for Characteristic Functions......Page p392.djvu
Notes......Page p397.djvu
11.1 Moments, Basic Properties......Page p400.djvu
11.2 Smoothness of a Characteristic Function and the Existence of Moments......Page p408.djvu
11.3 More about Smoothness of Characteristic Functions and Existence of Moments......Page p419.djvu
11.4 Absolute Moments......Page p428.djvu
11.5 Boundedness of the Spectra of Distribution Functions......Page p431.djvu
11.6 Integrable Characteristic Functions......Page p437.djvu
11.7 Analyticity of Distribution Functions......Page p439.djvu
11.8 Mean Concentration Function of a Distribution Function......Page p445.djvu
11.9 Some Properties of Analytic Characteristic Functions......Page p452.djvu
11.10 Characteristic Functions Analytic in the Half-Plane......Page p456.djvu
11.11 Entire Characteristic Functions I......Page p460.djvu
11.12 Entire Characteristic Functions II......Page p466.djvu
Notes......Page p473.djvu
12.1 Convergence of a Sequence of Random Variables......Page p475.djvu
12.2 The Borel Theorem......Page p486.djvu
12.3 The Zero-One Law......Page p488.djvu
12.4 The Equivalence Theorem......Page p492.djvu
12.5 The Three Series Theorem......Page p496.djvu
12.6 Sufficient Conditions for the Convergence of a Series......Page p501.djvu
12.7 Convergence Criteria and the Typical Function......Page p504.djvu
12.8 Rademacher and Steinhaus Functions......Page p507.djvu
12.9 Convergence of Products of Characteristic Functions......Page p513.djvu
12.10 Unconditional Convergence......Page p519.djvu
12.11 Absolute Convergence......Page p526.djvu
12.12 Essential Convergence......Page p530.djvu
Notes......Page p534.djvu
13.1 Continuity and Discontinuity Properties of the Sum of a Series......Page p536.djvu
13.2 Integrability of the Sum of a Series......Page p543.djvu
13.3 Magnitude of the Characteristic Functions of the Sums of Series......Page p546.djvu
13.4 Distribution Functions of the Sums of Rademacher Series; Characteristic Functions of Singular Distributions......Page p556.djvu
13.5 Further Theorems on Rademacher Series......Page p565.djvu
13.6 Sums of Independent Random Variables......Page p573.djvu
13.7 Convergent Systems......Page p583.djvu
13.8 Integrability of Sums of Series; Strong and Weak Convergences of Series......Page p586.djvu
13.9 Vanishing of the Sum of a Series......Page p590.djvu
13.10 Summability of Series......Page p596.djvu
Notes......Page p602.djvu
14.1 Fourier Series with Rademacher Coefficients......Page p605.djvu
14.2 Random Fourier Series......Page p612.djvu
14.3 Random Power Series, Convergence......Page p619.djvu
14.4 Convergence of Random Power Series with Identically and Independently Distributed Random Coefficients......Page p625.djvu
14.5 Analytic Continuation of Random Power Series......Page p628.djvu
14.6 Fourier Series with Orthogonal Random Coefficients......Page p634.djvu
Notes......Page p640.djvu
References......Page p642.djvu
Index......Page p661.djvu