Sums of Independent Random Variables

I. Probability Distributions and Characteristic Functions.- § 1. Random variables and probability distributions.- § 2. Characteristic functions.- § 3. Inversion formulae.- § 4. The convergence of sequences of distributions and characteristic functions.- § 5. Supplement.- II. Infinitely Divisible Distributions.- § 1. Definition and elementary properties of infinitely divisible distributions.- § 2. Canonical representation of infinitely divisible characteristic functions.- § 3. An auxiliary theorem.- § 4. Supplement.- III. Some Inequalities for the Distribution of Sums of Independent Random Variables.- § 1. Concentration functions.- § 2. Inequalities for the concentration functions of sums of independent random variables.- § 3. Inequalities for the distribution of the maximum of sums of independent random variables.- § 4. Exponential estimates for the distributions of sums of independent random variables.- § 5. Supplement.- IV. Theorems on Convergence to Infinitely Divisible Distributions.- § 1. Infinitely divisible distributions as limits of the distributions of sums of independent random variables.- § 2. Conditions for convergence to a given infinitely divisible distribution.- § 3. Limit distributions of class L and stable distributions.- § 4. The central limit theorem.- § 5. Supplement.- V. Estimates of the Distance Between the Distribution of a Sum of Independent Random Variables and the Normal Distribution.- § 1. Estimating the nearness of functions of bounded variation by the nearness of their Fourier-Stieltjes transforms.- § 2. The Esseen and Berry-Esseen inequalities.- § 3. Generalizations of Esseen's inequality.- § 4. Non-uniform estimates.- § 5. Supplement.- VI. Asymptotic Expansions in the Central Limit Theorem.- § 1. Formal construction of the expansions.- § 2 Auxiliary propositions.- § 3. Asymptotic expansions of the distribution function of a sum of independent identically distributed random variables.- § 4. Asymptotic expansions of the distribution function of a sum of independent non-identically distributed random variables, and of the derivatives of this function.- § 5. Supplement.- VII. Local Limit Theorems.- § 1. Local limit theorems for lattice distributions.- § 2. Local limit theorems for densities.- § 3. Asymptotic expansions in local limit theorems.- § 4. Supplement.- VIII. Probabilities of Large Deviations.- § 1. Introduction.- § 2. Asymptotic relations connected with Cramér's series.- § 3. Necessary and sufficient conditions for normal convergence in power zones.- § 4. Supplement.- IX. Laws of Large Numbers.- § 1. The weak law of large numbers.- § 2. Convergence of series of independent random variables.- § 3. The strong law of large numbers.- § 4. Convergence rates in the laws of large numbers.- § 5. Supplement.- X. The Law of the Iterated Logarithm.- § 1. Kolmogorov's theorem.- § 2. Generalization of Kolmogorov's theorem.- § 3. The central limit theorem and the law of the iterated logarithm.- § 4. Supplement.- Notes on Sources in the Literature.- References.- Subject Indes.- Table of Symbols and Abbreviations.