巴拿赫空间中的概率论

章节摘录

版权页：   The moment condition IE(‖X‖2/LL‖X‖) < ∞ is of course necessary inthis statement since it is not comparable to the tail behaviorlimt→t2P{‖X‖> t} = 0 necessary for the CLT. Despite this general sat-isfactory result, the question of the implication CLT → LIL is not solved forall that. Theorem 10.12 indicates that the spaces in which random variablessatisfying the CLT also satisfy the LIL are exactly those in which the CLTimplies the integrability property IE(‖X‖2/LL‖X‖) < eo. This is of coursethe case for cotype 2 spaces but the characterization of the CLT in Lp-spacesshows that Lp with p > 2 does not satisfy this property. An argument similarto the one used for Theorem 10.11, but this time with Theorem 9.16 insteadof Dvoretzky's theorem, then shows that the spaces satisfying CLT = LIL arenecessarily of cotype 2 + ε for every ε > 0. But a final characterization is stillto be obtained. 10.3 A Small Ball Criterion for the Central Limit Theorem In this last paragraph, we develop a criterion for the CLT which, while cer-tainly somewhat difficult to verify in practice, involves in its elaboration sev-eral interesting arguments and ideas developed throughout this book. Theresult therefore presents some interest from a theoretical point of view. Theidea of its proof can be used further for an almost sure randomized version ofthe CLT. Recall that we deal in all this chapter with a separable Banach space B.We noticed, prior to Theorem 3.3, that for a Gaussian Radon random variableG, with values in B each ball centered at the origin has a positive mass for thedistribution of G. Therefore, it follows that if X is a Borel random variablesatisfying the CLT in B, for every ε > 0.

内容概要

作者：（法国）李多科斯（Ledoux M.） （法国）Michel Talagrand

书籍目录

Introduction Notation Part 0. Isoperimetric Background and Generalities Chapter 1. Isoperimetric Inequalities and the Concentration of Measure Phenomenon 1.1 Some Isoperimetric Inequalities on the Sphere, in Gauss Space and on the Cube 1.2 An Isoperimetric Inequality for Product Measures 1.3 Martingale Inequalities Notes and References Chapter 2. Generalities on Banach Space Valued Random Variables and Random Processes 2.1 Banach Space Valued Radon Random Variables 2.2 Random Processes and Vector Valued Random Variables 2.3 Symmetric Random Variables and Lévy's Inequalities 2.4 Some Inequalities for Real Valued Random Variables Notes and References Part Ⅰ. Banach Space Valued Random Variables and Their Strong Limiting Properties Chapter 3. Gaussian Random Variables 3.1 Integrability and Tail Behavior 3.2 lntegrability of Gaussian Chaos 3.3 Comparison Theorems Notes and References Chapter 4. Rademacher Averages 4.1 Real Rademacher Averages 4.2 The Contraction Principle 4.3 Integrability and Tail Behavior of Rademacher Series 4.4 Integrability of Rademacher Chaos 4.5 Comparison Theorems Notes and References Chapter 5. Stable Random Variables 5.1 Representation of Stable Random Variables 5.2 Integrability and Tail Behavior 5.3 Comparison Theorems Notes and References Chapter 6. Sums of Independent Random Variables 6.1 Symmetrization and Some Inequalities for Sums of Independent Random Variables 6.2 Integrability of Sums of Independent Random Variables 6.3 Concentration and Tail Behavior Notes and References Chapter 7. The Strong Law of Large Numbers 7.1 A General Statement for Strong Limit Theorems 7.2 Examples of Laws of Large Numbers Notes and References Chapter 8. The Law of the Iterated Logarithm 8.1 Kolmogorov's Law of the Iterated Logarithm 8.2 Hartman-Wintner-Strassen's Law of the Iterated Logarithm 8.3 On the Identification of the Limits Notes and References Part Ⅱ. Tightness of Vector Valued Random Variables and Regularity of Random Processes Chapter 9. Type and Cotype of Banach Spaces 9.1 ιnp-Subspaces of Banach Spaces 9.2 Type and Cotype 9.3 Some Probabilistic Statements in Presence of Type and Cotype Notes and References Chapter 10. The Central Limit Theorem 10.1 Some General Facts About the Central Limit Theorem 10.2 Some Central Limit Theorems in Certain Banach Spaces 10.3 A Small Ball Criterion for the Central Limit Theorem Notes and References Chapter 11. Regularity of Random Processes 11.1 Regularity of Random Processes Under Metric Entropy Conditions 11.2 Regularity of Random Processes Under Majorizing Measure Conditions 11.3 Examples of Applications Notes and References Chapter 12. Regularity of Gaussian and Stable Processes 12.1 Regularity of Gaussian Processes 12.2 Necessary Conditions for the Boundedness and Continuity of Stable Processes 12.3 Applications and Conjectures on Rademacher Processes Notes and References Chapter 13. Stationary Processes and Random Fourier Series 13.1 Stationarity and Entropy 13.2 Random Fourier Series 13.3 Stable Random Fourier Series and Strongly Stationary Processes 13.4 Vector Valued Random Fourier Series Notes and References Chapter 14. Empirical Process Methods in Probability in Banach Spaces 14.1 The Central Limit Theorem for Lipschitz Processes 14.2 Empirical Processes and Random Geometry 14.3 Vapnik-Chervonenkis Classes of Sets Notes and References Chapter 15. Applications to Banach Space Theory 15.1 Subspaces of Small Codimension 15.2 Conjectures on Sudakov's Minoration for Chaos 15.3 An Inequality of J. Bourgain 15.4 Invertibility of Submatrices 15.5 Embedding Subspaces of Lp into ιNP 15.6 Majorizing Measures on Ellipsoids 15.7 Cotype of the Canonical Injection ιN∞→L2,1 15.8 Miscellaneous Problems Notes and References References Subject Index

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《巴拿赫空间中的概率论(英文)》由世界图书出版公司北京公司出版。

作者简介

《巴拿赫空间中的概率论(英文)》是一部全面讲述巴纳赫空间概率论的完美教程，作为概率论的一个分支，该理论已经得到了很好的发展。等周，测度和随机过程这些是学习巴纳赫空间概率论的基础技巧工具，书中全面介绍了巴纳赫空间中概率论的主要概念（积分，向量值随机变量的极限定理和随机变量的连续性）以及它们和巴纳赫空间几何的关系。《巴拿赫空间中的概率论(英文)》旨在从基础到重要结果将该理论的方方面面阐述清楚，测度和抽象随机过程技巧是《巴拿赫空间中的概率论(英文)》的重点，并且深入讨论了概率工具和经典巴纳赫理论。

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