Fourier Analysis in Probability Theory

PrefaceI Introduction 1.1 Measurable Space; Probability Space 1.2 Measurable Functions; Random Variables 1.3 Product Space 1.4 Integrals 1.5 The Fubini-Tonelli Theorem 1.6 Integrals on R1 1.7 Functions of Bounded Variation 1.8 Signed Measure; Decomposition Theorems 1.9 The Lebesgue Integral on R1 1.10 Inequalities 1.11 Convex Functions 1.12 Analytic Functions 1.13 Jensen's and Carleman's Theorems 1.14 Analytic Continuation 1.15 Maximum Modulus Theorem and Theorems of Phragmén-Lindelöf 1.16 Inner Product SpaceII Fourier Series and Fourier Transforms 2.1 The Riemann-Lebesgue Lemma 2.2 Fourier Series 2.3 The Fourier Transform of a Function in L1(—∞, ∞) 2.4 Magnitude of Fourier Coefficients; the Continuity Modulus 2.5 More About the Magnitude of Fourier Coefficients 2.6 Some Elementary Lemmas 2.7 Continuity and Magnitude of Fourier Transforms 2.8 Operations on Fourier Series 2.9 Operations on Fourier Transforms 2.10 Completeness of Trigonometric Functions 2.11 Unicity Theorem for Fourier Transforms 2.12 Fourier Series and Fourier Transform of Convolutions NotesIII Fourier-Stieltjes Coefficients, Fourier-Stieltjes Transforms and Characteristic Functions 3.1 Monotone Functions and Distribution Functions 3.2 Fourier-Stieltjes Series 3.3 Average of Fourier-Stieltjes Coefficients 3.4 Unicity Theorem for Fourier-Stieltjes Coefficients 3.5 Fourier-Stieltjes Transform and Characteristic Function 3.6 Periodic Characteristic Functions 3.7 Some Inequality Relations for Characteristic Functions 3.8 Average of a Characteristic Function 3.9 Convolution of Nondecreasing Functions 3.10 The Fourier-Stieltjes Transform of a Convolution and the Bernoulli Convolution NotesIV Convergence and Summability Theorems 4.1 Convergence of Fourier Series 4.2 Convergence of Fourier-Stieltjes Series 4.3 Fourier's Integral Theorems; Inversion Formulas for Fourier Transforms 4.4 Inversion Formula for Fourier-Stieltjes Transforms 4.5 Summability 4.6 (C,1)-Summability for Fourier Series 4.7 Abel-Summability for Fourier Series 4.8 Summability Theorems for Fourier Transforms 4.9 Determination of the Absolutely Continuous Component of a Nondecreasing Function 4.10 Fourier Series and Approximate Fourier Series of a Fourier-Stieltjes Transform 4.11 Some Examples, Using Fourier Transforms NotesV General Convergence Theorems 5.1 Nature of the Problems 5.2 Some General Convergence Theorems I 5.3 Some General Convergence Theorems II 5.4 General Convergence Theorems for the Stieltjes Integral 5.5 Wiener's Formula 5.6 Applications of General Convergence Theorems to the Estimates of a Distribution Function NotesVI L2-Theory of Fourier Series and Fourier Transforms 6.1 Fourier Series in an Inner Product Space 6.2 Fourier Transform of a Function in L2(-∞, ∞) 6.3 The Class H2 of Analytic Functions 6.4 A Theorem of Szegö and Smirnov 6.5 The Class ƃ2 of Analytic Functions 6.6 A Theorem of Paley and Wiener NotesVII Laplace and Mellin Transforms 7.1 The Laplace Transform 7.2 The Convergence Abscissa 7.3 Analyticity of a Laplace-Stieltjes Transform 7.4 Inversion Formulas for Laplace Transforms 7.5 The Laplace Transform of a Convolution 7.6 Operations of Laplace Transforms and Some Examples 7.7 The Bilateral Laplace-Stieltjes Transform 7.8 Mellin-Stieltjes Transforms 7.9 The Mellin Transform NotesVIII More Theorems on Fourier and Laplace Transforms 8.1 A Theorem of Hardy 8.2 A Theorem of Paley and Wiener on Exponential Entire Functions 8.3 Theorems of Ingham and Levinson 8.4 Singularities of Laplace Transforms 8.5 Abelian Theorems for Laplace Transforms 8.6 Tauberian Theorems 8.7 Multiple Fourier Series and Transforms 8.8 Nondecreasing Functions and Distribution Functions in Rm 8.9 The Multiple Fourier-Stieltjes Transform NotesIX Convergence of Distribution Functions and Characteristic Functions 9.1 Helly Theorems and Convergence of Nondecreasing Functions 9.2 Convergence of Distribution Functions with Bounded Spectra 9.3 Convergence of Distribution Functions 9.4 Continuous Distribution Functions: A General Integral Transform of a Characteristic Function 9.5 A Basic Theorem on Analytic Characteristic Functions 9.6 Continuity Theorems on Intervals and Uniqueness Theorems 9.7 The Compact Set of Characteristic Functions NotesX Some Properties of Characteristic Functions 10.1 Characteristic Properties of Fourier Coefficients 10.2 Basic Theorems on Characterization of a Characteristic Function 103. Characteristic Properties of Characteristic Functions 10.4 Functions of the Wiener Class 10.5 Some Sufficient Criteria for Characteristic Functions 10.6 More Criteria for Characteristic Functions NotesXI Distribution Functions and their Characteristic Functions 11.1 Moments, Basic Properties 11.2 Smoothness of a Characteristic Function and the Existence of Moments 11.3 More About Smoothness of Characteristic Functions and Existence of Moments 11.4 Absolute Moments 11.5 Boundedness of the Spectra of Distribution Functions 11.6 Integrable Characteristic Functions 11.7 Analyticity of Distribution Functions 11.8 Mean Concentration Function of a Distribution Function 11.9 Some Properties of Analytic Characteristic Functions 11.10 Characteristic Functions Analytic in the Half-Plane 11.11 Entire Characteristic Functions I 11.12 Entire Characteristic Functions II NotesXII Convergence of Series of Independent Random Variables 12.1 Convergence of a Sequence of Random Variables 12.2 The Borel Theorem 12.3 The Zero-One Law 12.4 The Equivalence Theorem 12.5 The Three Series Theorem 12.6 Sufficient Conditions for the Convergence of a Series 12.7 Convergence Criteria and the Typical Function 12.8 Rademacher and Steinhaus Functions 12.9 Convergence of Products of Characteristic Functions 12.10 Unconditional Convergence 12.11 Absolute Convergence 12.12 Essential Convergence NotesXIII Properties of Sums of Independent Random Variables; Convergence of Series in the Mean 13.1 Continuity and Discontinuity Properties of the Sum of a Series 13.2 Integrability of the Sum of a Series 13.3 Magnitude of the Characteristic Functions of the Sums of Series 13.4 Distribution Functions of the Sums of Rademacher Series; Characteristic Functions of Singular Distributions 13.5 Further Theorems on Rademacher Series 13.6 Sums of Independent Random Variables 13.7 Convergent Systems 13.8 Integrability of Sums of Series; Strong and Weak Convergences of Series 13.9 Vanishing of the Sum of a Series 13.10 Summability of Series NotesXIV Some Special Series of Random Variables 14.1 Fourier Series with Rademacher Coefficients 14.2 Random Fourier Series 14.3 Random Power Series, Convergence 14.4 Convergence of Random Power Series with Identically and Independently Distributed Random Coefficients 14.5 Analytic Continuation of Random Power Series 14.6 Fourier Series with Orthogonal Random Coefficients NotesReferencesIndex