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El. knyga: Uniform Central Limit Theorems

(Massachusetts Institute of Technology)
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In this new edition of a classic work on empirical processes the author, an acknowledged expert, gives a thorough treatment of the subject with the addition of several proved theorems not included in the first edition, including the BretagnolleMassart theorem giving constants in the KomlosMajorTusnady rate of convergence for the classical empirical process, Massart's form of the DvoretzkyKieferWolfowitz inequality with precise constant, Talagrand's generic chaining approach to boundedness of Gaussian processes, a characterization of uniform GlivenkoCantelli classes of functions, Giné and Zinn's characterization of uniform Donsker classes, and the BousquetKoltchinskiiPanchenko theorem that the convex hull of a uniform Donsker class is uniform Donsker. The book will be an essential reference for mathematicians working in infinite-dimensional central limit theorems, mathematical statisticians, and computer scientists working in computer learning theory. Problems are included at the end of each chapter so the book can also be used as an advanced text.

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This expanded edition of the classic work on empirical processes now boasts several new proved theorems not in the first.
Preface to the Second Edition xi
1 Donsker's Theorem and Inequalities
1(60)
1.1 Empirical Processes: The Classical Case
6(1)
1.2 Metric Entropy and Capacity
7(2)
1.3 Inequalities
9(6)
1.4 Proof of the Bretagnolle-Massart Theorem
15(24)
1.5 The DKW Inequality in Massart's Form
39(22)
2 Gaussian Processes; Sample Continuity
61(72)
2.1 General Empirical and Gaussian Processes
61(1)
2.2 Some Definitions
62(5)
2.3 Bounds for Gaussian Vectors
67(6)
2.4 Inequalities for Gaussian Distributions
73(9)
2.5 Sample Boundedness
82(3)
2.6 Gaussian Measures and Convexity
85(3)
2.7 Regularity of the Isonormal Process
88(6)
2.8 A Metric Entropy Condition for Continuity
94(6)
2.9 Gaussian Concentration Inequalities
100(8)
2.10 Generic Chaining
108(9)
2.11 Homogeneous and Quasi-Homogeneous Sets in H
117(4)
2.12 Sample Continuity and Compactness
121(4)
2.13 Two-Series and One-Series Theorems
125(8)
3 Definition of Donsker Classes
133(42)
3.1 Definitions: Convergence in Law
133(4)
3.2 Measurable Cover Functions
137(6)
3.3 Almost Uniform, Outer Probability Convergence
143(2)
3.4 Perfect Functions
145(4)
3.5 Almost Surely Convergent Realizations
149(5)
3.6 Conditions Equivalent to Convergence in Law
154(5)
3.7 Asymptotic Equicontinuity
159(3)
3.8 Unions of Donsker Classes
162(1)
3.9 Sequences of Sets and Functions
163(5)
3.10 Donsker Classes and Sequential Limits
168(1)
3.11 Convex Hulls of Donsker Classes
168(7)
4 Vapnik---Cervonenkis Combinatorics
175(38)
4.1 Vapnik---Cervonenkis Classes of Sets
175(4)
4.2 Generating Vapnik---Cervonenkis Classes
179(4)
4.3 Maximal Classes
183(2)
4.4 Classes of Index 1
185(7)
4.5 Combining VC Classes
192(8)
4.6 Probability Laws and Independence
200(4)
4.7 VC Properties of Function Classes
204(1)
4.8 Classes of Functions and Dual Density
205(8)
5 Measurability
213(26)
5.1 Sufficiency
215(7)
5.2 Admissibility
222(7)
5.3 Suslin Properties and Selection
229(10)
6 Limit Theorems for VC-Type Classes
239(30)
6.1 Glivenko---Cantelli Theorems
239(8)
6.2 Glivenko---Cantelli Properties for Given P
247(4)
6.3 Pollard's Central Limit Theorem
251(9)
6.4 Necessary Conditions for Limit Theorems
260(9)
7 Metric Entropy with Bracketing
269(15)
7.1 The Blum-DeHardt Theorem
269(5)
7.2 Bracketing Central Limit Theorems
274(5)
7.3 The Power Set of a Countable Set
279(5)
8 Approximation of Functions and Sets
284(35)
8.1 Introduction: The Hausdorff Metric
284(3)
8.2 Spaces of Differentiable Functions and Sets
287(13)
8.3 Lower Layers
300(5)
8.4 Metric Entropy of Classes of Convex Sets
305(14)
9 Two Samples and the Bootstrap
319(29)
9.1 The Two-Sample Case
319(4)
9.2 A Bootstrap CLT
323(22)
9.3 Other Aspects of the Bootstrap
345(3)
10 Uniform and Universal Limit Theorems
348(43)
10.1 Uniform Glivenko---Cantelli Classes
348(12)
10.2 Universal Donsker Classes
360(6)
10.3 Metric Entropy of Convex Hulls in Hilbert Space
366(6)
10.4 Uniform Donsker Classes
372(16)
10.5 Universal Glivenko---Cantelli Classes
388(3)
11 Classes Too Large to Be Donsker
391(26)
11.1 Universal Lower Bounds
391(2)
11.2 An Upper Bound
393(2)
11.3 Poissonization and Random Sets
395(5)
11.4 Lower Bounds in Borderline Cases
400(10)
11.5 Proof of Theorem 11.10
410(7)
Appendices
A Differentiating under an Integral Sign
417(7)
B Multinomial Distributions
424(3)
C Measures on Nonseparable Metric Spaces
427(3)
D An Extension of Lusin's Theorem
430(2)
E Bochner and Pettis Integrals
432(5)
F Nonexistence of Some Linear Forms
437(3)
G Separation of Analytic Sets
440(3)
H Young---Orlicz Spaces
443(3)
I Versions of Isonormal Processes
446(3)
Bibliography 449(14)
Notation Index 463(2)
Author Index 465(3)
Subject Index 468
R. M. Dudley is a Professor of Mathematics at the Massachusetts Institute of Technology in Cambridge, Massachusetts.
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