Sums of Independent Random Variables [Paperback]

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 Esseens 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 funcl³(