Sums of Independent Random Variables

ISBN-13: 9783642658112 / Angielski / Miękka / 2011 / 348 str.

V.V. Petrov;A.A. Brown

Opis

The classic "Limit Dislribntions fOT slt1ns of Independent Ramdorn Vari- ables" by B. V. Gnedenko and A. N. Kolmogorov was published in 1949. Since then the theory of summation of independent variables has devel- oped rapidly. Today a summing-up of the studies in this area, and their results, would require many volumes. The monograph by I. A. Ibragi- mov and Yu. V. I~innik, "Independent and Stationarily Connected VaTiables", which appeared in 1965, contains an exposition of the contem- porary state of the theory of the summation of independent identically distributed random variables. The present book borders on that of Ibragimov and Linnik, sharing only a few common areas. Its main focus is on sums of independent but not necessarily identically distri- buted random variables. It nevertheless includes a number of the most recent results relating to sums of independent and identically distributed variables. Together with limit theorems, it presents many probahilistic inequalities for sums of an arbitrary number of independent variables. The last two chapters deal with the laws of large numbers and the law of the iterated logarithm. These questions were not treated in Ibragimov and Linnik; Gnedenko and KolmogoTOv deals only with theorems on the weak law of large numbers. Thus this book may be taken as complementary to the book by Ibragimov and Linnik. I do not, however, assume that the reader is familiar with the latter, nor with the monograph by Gnedenko and Kolmogorov, which has long since become a bibliographical rarity.

Spis treści

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.