Contributions to Probability

List of ContributorsPrefaceEugene LukacsPart I. ProbabilityProbability Theory Lagrange's Theorem and Thin Subsequences of Squares References A Kind of Random (= Stochastic) Integral 0. Random Functions 1. Measurable Random Functions 2. Random ( = Stochastic) Measure 3. Random Integral References Continuation of Distribution Functions 1. Introduction 2. The Normal Distribution Function 3. Analytic Distribution Functions 4. Continuation Theorems for Special Classes of i.d. d.f. 5. Uniqueness of Symmetric Distribution Functions References The Arc Sine Law of Paul Lévy 1. Introduction 2. The Coin Tossing Game 3. Paul Lévy's Arc Sine Law 4. An Alternative Approach 5. Exact Distributions 6. Notes 7. Lévy's Heuristic Method 8. The Case of Independent Sequences 9. The Case of Dependent Sequences ReferencesLimit Theorems General Limit Theorems for Products with Applications to Convolution Products of Measures 1. Introduction 2. Some Fundamental Identities and Inequalities 3. Convergence of Powers in Seminorms to Infinitely Divisible Elements 4. Convergence of Products in Seminorms References Stable Limit Law and Weak Law of Large Numbers for Hilbert Space with "Large-O" Rates 1. Introduction 2. Notations and Preliminaries 3. Two General Large-O Approximation Theorems 4. Stable Limit Law on H with Rates 5. The Central Limit Theorem on H 6. The Weak Law of Large Numbers 7. Limit Theorems for Random Vectors in Rm References The Arithmetic of Distribution Functions 1. Introduction 2. The Class L 3. Infinitely Divisible Characteristic Functions with Absolutely Continuous Spectral Functions 4. Products of Poisson-Type Characteristic Functions 5. Independent Sets 6. Infinitely Divisible Characteristic Functions with Continuous Spectral Functions 7. Indecomposable Laws 8. Indecomposable Factors 9. a Decompositions References On the Tails of a Class of Infinitely Divisible Distributions 1. Introduction 2. Some Lemmas 3. Proof of Theorem 1.2 4. On Another Theorem of Elliott and Erdös References Fourier Transform Methods in the Study of Limit Theorems in a Hilbert Space 1. Introduction 2. Preliminaries 3. Infinitely Divisible Probability Measures 4. The General Central Limit Problem in a Hilbert Space 5. Self-Decomposable and Stable Measures in H 6. Semi-Stable Measures in H References Polynomials in Gaussian Variables and Infinite Divisibility? References On the Nondifferentiability of the Wiener Sheet 1. Introduction 2. The Nondifferentiability of the Wiener Sheet in the Direction of the X Axis 3. The Wiener Sheet Is Nowhere Differentiable in Any Direction References The Degree of Vertices on a Randomly Growing Tree 1. Introduction 2. The State-Homogeneous Case 3. A Simple Practical Example References Uniform Convergence of Random Trigonometric Series and Sample Continuity of Weakly Stationary Processes 1. Introduction 2. Sequences of Random Variables Satisfying the Condition Mr 3. Uniform Convergence of Random Trigonometric Series 4. Weakly Stationary Process of Class Mr: Approximate Fourier Series 5. Sample Continuity of a Weakly Stationary Process of Mr References Stochastic Equations Driven by Random Measures and Semimartingales Introduction Notations 1. Some Preliminaries on Random Measures 2.