Sums of Independent Random Variables
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| Format: | eBook |
| Language: | English |
| Published: |
Berlin, Heidelberg
Springer Berlin Heidelberg
1975, 1975
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| Edition: | 1st ed. 1975 |
| Series: | Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge, A Series of Modern Surveys in Mathematics
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 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. Formalconstruction of the expansions
- § 2 Auxiliary propositions
- 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
- § 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