Statistical Learning Theory,9780471030034

Statistical Learning Theory

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Edition: 1st
ISBN13: 9780471030034
ISBN10: 0471030031
Format: Hardcover
Pub. Date: 1998-09-30
Publisher(s): Wiley-Interscience
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Summary

A comprehensive look at learning and generalization theory. The statistical theory of learning and generalization concerns the problem of choosing desired functions on the basis of empirical data. Highly applicable to a variety of computer science and robotics fields, this book offers lucid coverage of the theory as a whole. Presenting a method for determining the necessary and sufficient conditions for consistency of learning process, the author covers function estimates from small data pools, applying these estimations to real-life problems, and much more.

Author Biography

Vladimir Naumovich Vapnik is one of the main developers of the Vapnik-Chervonenkis theory of statistical learning, and the co-inventor of the support vector machine method, and support vector clustering algorithm.

Table of Contents

PREFACE xxi
Introduction: The Problem of Induction and Statistical Inference 1(18)
0.1 Learning Paradigm in Statistics 1(1)
0.2 Two Approaches to Statistical Inference: Particular (Parametric Inference) and General (Nonparametric Inference) 2(2)
0.3 The Paradigm Created by the Parametric Approach 4(1)
0.4 Shortcoming of the Parametric Paradigm 5(1)
0.5 After the Classical Paradigm 6(1)
0.6 The Renaissance 7(1)
0.7 The Generalization of the Glivenko-Cantelli-Kolmogorov Theory 8(2)
0.8 The Structural Risk Minimization Principle 10(1)
0.9 The Main Principle of Inference from a Small Sample Size 11(2)
0.10 What This Book is About 13(6)
I THEORY OF LEARNING AND GENERALIZATION 19(356)
1 Two Approaches to the Learning Problem
19(40)
1.1 General Model of Learning from Examples
19(2)
1.2 The Problem of Minimizing the Risk Functional from Empirical Data
21(3)
1.3 The Problem of Pattern Recognition
24(2)
1.4 The Problem of Regression Estimation
26(2)
1.5 Problem of Interpreting Results of Indirect Measuring
28(2)
1.6 The Problem of Density Estimation (the Fisher-Wald Setting)
30(2)
1.7 Induction Principles for Minimizing the Risk Functional on the Basis of Empirical Data
32(1)
1.8 Classical Methods for Solving the Function Estimation Problems
33(2)
1.9 Identification of Stochastic Objects: Estimation of the Densities and Conditional Densities
35(3)
1.9.1 Problem of Density Estimation. Direct Setting
35(1)
1.9.2 Problem of Conditional Probability Estimation
36(1)
1.9.3 Problem of Conditional Density Estimation
37(1)
1.10 The Problem of Solving an Approximately Determined Integral Equation
38(1)
1.11 Glivenko-Cantelli Theorem
39(5)
1.11.1 Convergence in Probability and Almost Sure Convergence
40(2)
1.11.2 Glivenko-Cantelli Theorem
42(1)
1.11.3 Three Important Statistical Laws
42(2)
1.12 Ill-Posed Problems
44(4)
1.13 The Structure of the Learning Theory
48(3)
Appendix to Chapter 1: Methods for Solving III-Posed Problems
51(8)
A1.1 The Problem of Solving an Operator Equation
51(2)
A1.2 Problems Well-Posed in Tikhonov's Sense
53(1)
A1.3 The Regularization Method
54(1)
A1.3.1 Idea of Regularization Method
54(1)
A1.3.2 Main Theorems About the Regularization Method
55(4)
2 Estimation of the Probability Measure and Problem of Learning
59(20)
2.1 Probability Model of a Random Experiment
59(2)
2.2 The Basic Problem of Statistics
61(4)
2.2.1 The Basic Problems of Probability and Statistics
61(1)
2.2.2 Uniform Convergence of Probability Measure Estimates
62(3)
2.3 Conditions for the Uniform Convergence of Estimates to the Unknown Probability Measure
65(4)
2.3.1 Structure of Distribution Function
65(3)
2.3.2 Estimator that Provides Uniform Convergence
68(1)
2.4 Partial Uniform Convergence and Generalization of Glivenko-Cantelli Theorem
69(3)
2.4.1 Definition of Partial Uniform Convergence
69(2)
2.4.2 Generalization of the Glivenko-Cantelli Problem
71(1)
2.5 Minimizing the Risk Functional Under the Condition of Uniform Convergence of Probability Measure Estimates
72(2)
2.6 Minimizing the Risk Functional Under the Condition of Partial Uniform Convergence of Probability Measure Estimates
74(3)
2.7 Remarks About Modes of Convergence of the Probability Measure Estimates and Statements of the Learning Problem
77(2)
3 Conditions for Consistency of Empirical Risk Minimization Principle
79(42)
3.1 Classical Definition of Consistency
79(3)
3.2 Definition of Strict (Nontrivial) Consistency
82(3)
3.2.1 Definition of Strict Consistency for the Pattern Recognition and the Regression Estimation Problems
82(2)
3.2.2 Definition of Strict Consistency for the Density Estimation Problem
84(1)
3.3 Empirical Processes
85(3)
3.3.1 Remark on the Law of Large Numbers and Its Generalization
86(2)
3.4 The Key Theorem of Learning Theory (Theorem About Equivalence)
88(1)
3.5 Proof of the Key Theorem
89(3)
3.6 Strict Consistency of the Maximum Likelihood Method
92(1)
3.7 Necessary and Sufficient Conditions for Uniform Convergence of Frequencies to Their Probabilities
93(5)
3.7.1 Three Cases of Uniform Convergence
93(1)
3.7.2 Conditions of Uniform Convergence in the Simplest Model
94(1)
3.7.3 Entropy of a Set of Functions
95(2)
3.7.4 Theorem About Uniform Two-Sided Convergence
97(1)
3.8 Necessary and Sufficient Conditions for Uniform Convergence of Means to Their Expectations for a Set of Real-Valued Bounded Functions
98(2)
3.8.1 Entropy of a Set of Real-Valued Functions
98(1)
3.8.2 Theorem About Uniform Two-Sided Convergence
99(1)
3.9 Necessary and Sufficient Conditions for Uniform Convergence of Means to Their Expectations for Sets of Unbounded Functions
100(6)
3.9.1 Proof of Theorem 3.5
101(5)
3.10 Kant's Problem of Demarcation and Popper's Theory of Nonfalsifiability
106(2)
3.11 Theorems About Nonfalsifiability
108(4)
3.11.1 Case of Complete Nonfalsifiability
108(1)
3.11.2 Theorem About Partial Nonfalsifiability
109(1)
3.11.3 Theorem About Potential Nonfalsifiability
110(2)
3.12 Conditions for One-Sided Uniform Convergence and Consistency of the Empirical Risk Minimization Principle
112(6)
3.13 Three Milestones in Learning Theory
118(3)
4 Bounds on the Risk for Indicator Loss Functions
121(62)
4.1 Bounds for the Simplest Model: Pessimistic Case
122(3)
4.1.1 The Simplest Model
123(2)
4.2 Bounds for the Simplest Model: Optimistic Case
125(2)
4.3 Bounds for the Simplest Model: General Case
127(2)
4.4 The Basic Inequalities: Pessimistic Case
129(2)
4.5 Proof of Theorem 4.1
131(6)
4.5.1 The Basic Lemma
131(1)
4.5.2 Proof of Basic Lemma
132(2)
4.5.3 The Idea of Proving Theorem 4.1
134(1)
4.5.4 Proof of Theorem 4.1
135(2)
4.6 Basic Inequalities: General Case
137(2)
4.7 Proof of Theorem 4.2
139(5)
4.8 Main Nonconstructive Bounds
144(1)
4.9 VC Dimension
145(5)
4.9.1 The Structure of the Growth Function
145(3)
4.9.2 Constructive Distribution-Free Bounds on Generalization Ability
148(1)
4.9.3 Solution of Generalized Glivenko-Cantelli Problem
149(1)
4.10 Proof of Theorem 4.3
150(5)
4.11 Example of the VC Dimension of the Different Sets of Functions
155(5)
4.12 Remarks About the Bounds on the Generalization Ability of Learning Machines
160(3)
4.13 Bound on Deviation of Frequencies in Two Half-Samples
163(6)
Appendix to Chapter 4: Lower Bounds on the Risk of the ERM Principle
169(14)
A4.1 Two Strategies in Statistical Inference
169(2)
A4.2 Minimax Loss Strategy for Learning Problems
171(2)
A4.3 Upper Bounds on the Maximal Loss for the Empirical Risk Minimization Principle
173(1)
A4.3.1 Optimistic Case
173(1)
A4.3.2 Pessimistic Case
174(3)
A4.4 Lower Bound for the Minimax Loss Strategy in the Optimistic Case
177(2)
A4.5 Lower Bound for Minimax Loss Strategy for the Pessimistic Case
179(4)
5 Bounds on the Risk for Real-Valued Loss Functions
183(36)
5.1 Bounds for the Simplest Model: Pessimistic Case
183(3)
5.2 Concepts of Capacity for the Sets of Real-Valued Functions
186(6)
5.2.1 Nonconstructive Bounds on Generalization for Sets of Real-Valued Functions
186(2)
5.2.2 The Main Idea
188(2)
5.2.3 Concepts of Capacity for the Set of Real-Valued Functions
190(2)
5.3 Bounds for the General Model: Pessimistic Case
192(2)
5.4 The Basic Inequality
194(2)
5.4.1 Proof of Theorem 5.2
195(1)
5.5 Bounds for the General Model: Universal Case
196(4)
5.5.1 Proof of Theorem 5.3
198(2)
5.6 Bounds for Uniform Relative Convergence
200(7)
5.6.1 Proof of Theorem 5.4 for the Case p greater than 2
201(3)
5.6.2 Proof of Theorem 5.4 for the Case 1 less than p is greater than equal to 2
204(3)
5.7 Prior Information for the Risk Minimization Problem in Sets of Unbounded Loss Functions
207(3)
5.8 Bounds on the Risk for Sets of Unbounded Nonnegative Functions
210(4)
5.9 Sample Selection and the Problem of Outliers
214(2)
5.10 The Main Results of the Theory of Bounds
216(3)
6 The Structural Risk Minimization Principle
219(74)
6.1 The Scheme of the Structural Risk Minimization Induction Principle
219(5)
6.1.1 Principle of Structural Risk Minimization
221(3)
6.2 Minimum Description Length and Structural Risk Minimization Inductive Principles
224(5)
6.2.1 The Idea About the Nature of Random Phenomena
224(1)
6.2.2 Minimum Description Length Principle for the Pattern Recognition Problem
224(2)
6.2.3 Bounds for the Minimum Description Length Principle
226(1)
6.2.4 Structural Risk Minimization for the Simplest Model and Minimum Description Length Principle
227(1)
6.2.5 The Shortcoming of the Minimum Description Length Principle
228(1)
6.3 Consistency of the Structural Risk Minimization Principle and Asymptotic Bounds on the Rate of Convergence
229(8)
6.3.1 Proof of the Theorems
232(3)
6.3.2 Discussions and Example
235(2)
6.4 Bounds for the Regression Estimation Problem
237(9)
6.4.1 The Model of Regression Estimation by Series Expansion
238(3)
6.4.2 Proof of Theorem 6.4
241(5)
6.5 The Problem of Approximating Functions
246(11)
6.5.1 Three Theorems of Classical Approximation Theory
248(3)
6.5.2 Curse of Dimensionality in Approximation Theory
251(1)
6.5.3 Problem of Approximation in Learning Theory
252(2)
6.5.4 The VC Dimension in Approximation Theory
254(3)
6.6 Problem of Local Risk Minimization
257(14)
6.6.1 Local Risk Minimization Model
359(3)
6.6.2 Bounds for the Local Risk Minimization Estimator
262(3)
6.6.3 Proofs of the Theorems
265(3)
6.6.4 Structural Risk Minimization Principle for Local Function Estimation
268(3)
Appendix to Chapter 6: Estimating Functions on the Basis of Indirect Measurements
271(22)
A6.1 Problems of Estimating the Results of Indirect Measurements
271(2)
A6.2 Theorems on Estimating Functions Using Indirect Measurements
273(3)
A6.3 Proofs of the Theorems
276(1)
A6.3.1 Proofs of Theorem A6.1
276(5)
A6.3.2 Proofs of Theorem A6.2
281(2)
A6.3.3 Proof of Problem A6.3
283(10)
7 Stochastic III-Posed Problems
293(46)
7.1 Stochastic III-Posed Problems
293(4)
7.2 Regularization Method for Solving Stochastic III-Posed Problems
297(2)
7.3 Proofs of the Theorems
299(6)
7.3.1 Proof of Theorem 7.1
299(3)
7.3.2 Proof of Theorem 7.2
302(1)
7.3.3 Proof of Theorem 7.3
303(2)
7.4 Conditions for Consistency of the Methods of Density Estimation
305(3)
7.5 Nonparametric Estimators of Density: Estimators Based on Approximations of the Distribution Function by an Empirical Distribution Function
308(7)
7.5.1 The Parzen Estimators
308(5)
7.5.2 Projection Estimators
313(1)
7.5.3 Splines Estimate of the Density. Approximation by Splines of the Odd Order
313(1)
7.5.4 Spline Estimate of the Density. Approximation by Splines of the Even Order
314(1)
7.6 Nonclassical Estimators
315(4)
7.6.1 Estimators for the Distribution Function
315(1)
7.6.2 Polygon Approximation of Distribution Function
316(1)
7.6.3 Kernel Density Estimator
316(2)
7.6.4 Projection Method of the Density Estimator
318(1)
7.7 Asymptotic Rate of Convergence for Smooth Density Functions
319(3)
7.8 Proof of Theorem 7.4
322(5)
7.9 Choosing a Value of Smoothing (Regularization) Parameter for the Problem of Density Estimation
327(3)
7.10 Estimation of the Ratio of Two Densities
330(4)
7.10.1 Estimation of Conditional Densities
333(1)
7.11 Estimation of Ratio of Two Densities on the Line
334(3)
7.12 Estimation of a Conditional Probability on a Line
337(2)
8 Estimating the Values of Function at Given Points
339(36)
8.1 The Scheme of Minimizing the Overall Risk
339(4)
8.2 The Method of Structural Minimization of the Overall Risk
343(1)
8.3 Bounds on the Uniform Relative Deviation of Frequencies in Two Subsamples
344(3)
8.4 A Bound on the Uniform Relative Deviation of Means in Two Subsamples
347(3)
8.5 Estimation of Values of an Indicator Function in a Class of Linear Decision Rules
350(5)
8.6 Sample Selection for Estimating the Values of an Indicator Function
355(4)
8.7 Estimation of Values of a Real Function in the Class of Functions Linear in Their Parameters
359(3)
8.8 Sample Selection for Estimation of Values of Real-Valued Functions
362(1)
8.9 Local Algorithms for Estimating Values of an Indicator Function
363(2)
8.10 Local Algorithms for Estimating Values of a Real-Valued Function
365(2)
8.11 The Problem of Finding the Best Point in a Given Set
367(8)
8.11.1 Choice of the Most Probable Representative of the First Class
368(2)
8.11.2 Choice of the Best Point of a Given Set
370(5)
II SUPPORT VECTOR ESTIMATION OF FUNCTIONS 375(196)
9 Perceptrons and Their Generalizations
375(26)
9.1 Rosenblatt's Perceptron
375(5)
9.2 Proofs of the Theorems
380(3)
9.2.1 Proof of Novikoff Theorem
380(2)
9.2.2 Proof of Theorem 9.3
382(1)
9.3 Method of Stochastic Approximation and Sigmoid Approximation of Indicator Functions
383(4)
9.3.1 Method of Stochastic Approximation
384(1)
9.3.2 Sigmoid Approximations of Indicator Functions
385(2)
9.4 Method of Potential Functions and Radial Basis Functions
387(3)
9.4.1 Method of Potential Functions in Asymptotic Learning Theory
388(1)
9.4.2 Radial Basis Function Method
389(1)
9.5 Three Theorems of Optimization Theory
390(5)
9.5.1 Fermat's Theorem (1629)
390(1)
9.5.2 Lagrange Multipliers Rule (1788)
391(2)
9.5.3 Kuhn-Tucker Theorem (1951)
393(2)
9.6 Neural Networks
395(6)
9.6.1 The Back-Propagation Method
395(3)
9.6.2 The Back-Propagation Algorithm
398(1)
9.6.3 Neural Networks for the Regression Estimation Problem
399(1)
9.6.4 Remarks on the Back-Propagation Method
399(2)
10 The Support Vector Method for Estimating Indicator Functions
401(42)
10.1 The Optimal Hyperplane
401(7)
10.2 The Optimal Hyperplane for Nonseparable Sets
408(4)
10.2.1 The Hard Margin Generalization of the Optimal Hyperplane
408(3)
10.2.2 The Basic Solution. Soft Margin Generalization
411(1)
10.3 Statistical Properties of the Optimal Hyperplane
412(3)
10.4 Proofs of the Theorems
415(6)
10.4.1 Proof of Theorem 10.3
415(1)
10.4.2 Proof of Theorem 10.4
415(1)
10.4.3 Leave-One-Out Procedure
416(1)
10.4.4 Proof of Theorem 10.5 and Theorem 9.2
417(1)
10.4.5 Proof of Theorem 10.6
418(3)
10.4.6 Proof of Theorem 10.7
421(1)
10.5 The Idea of the Support Vector Machine
421(5)
10.5.1 Generalization in High-Dimensional Space
422(1)
10.5.2 Hilbert-Schmidt Theory and Mercer Theorem
423(1)
10.5.3 Constructing SV Machines
424(2)
10.6 One More Approach to the Support Vectors Method
426(2)
10.6.1 Minimizing the Number of Support Vectors
426(1)
10.6.2 Generalization for the Nonseparable Case
427(1)
10.6.3 Linear Optimization Method for SV Machines
427(1)
10.7 Selection of SV Machine Using Bounds
428(2)
10.8 Examples of SV Machines for Pattern Recognition
430(4)
10.8.1 Polynomial Support Vector Machines
430(1)
10.8.2 Radial Basis Function SV Machines
431(1)
10.8.3 Two-Layer Neural SV Machines
432(2)
10.9 Support Vector Method for Transductive Inference
434(3)
10.10 Multiclass Classification
437(3)
10.11 Remarks on Generalization of the SV Method
440(3)
11 The Support Vector Method for Estimating Real-Valued Functions
443(50)
11.1 Epsilon-Insensitive Loss Functions
443(2)
11.2 Loss Functions for Robust Estimators
445(3)
11.3 Minimizing the Risk with Epsilon-Insensitive Loss Functions
448(6)
11.3.1 Minimizing the Risk for a Fixed Element of the Structure
449(3)
11.3.2 The Basic Solutions
452(1)
11.3.3 Solution for the Huber Loss Function
453(1)
11.4 SV Machines for Function Estimation
454(6)
11.4.1 Minimizing the Risk for a Fixed Element of the Structure in Feature Space
455(1)
11.4.2 The Basic Solutions in Feature Space
456(2)
11.4.3 Solution for Huber Loss Function in Feature Space
458(1)
11.4.4 Linear Optimization Method
459(1)
11.4.5 Multi-Kernel Decomposition of Functions
459(1)
11.5 Constructing Kernels for Estimation of Real-Valued Functions
460(4)
11.5.1 Kernels Generating Expansion on Polynomials
461(1)
11.5.2 Constructing Multidimensional Kernels
462(2)
11.6 Kernels Generating Splines
464(4)
11.6.1 Spline of Order d with a Finite Number of Knots
464(1)
11.6.2 Kernels Generating Splines with an Infinite Number of Knots
465(1)
11.6.3 B(d) Spline Approximations
466(2)
11.6.4 B(d) Splines with an Infinite Number of Knots
468(1)
11.7 Kernels Generating Fourier Expansions
468(3)
11.7.1 Kernels for Regularized Fourier Expansions
469(2)
11.8 The Support Vector ANOVA Decomposition (SVAD) for Function Approximation and Regression Estimation
471(2)
11.9 SV Method for Solving Linear Operator Equations
473(6)
11.9.1 The SV Method
473(5)
11.9.2 Regularization by Choosing Parameters of Epsilon(i) Insensitivity
478(1)
11.10 SV Method of Density Estimation
479(5)
11.10.1 Spline Approximation of a Density
480(1)
11.10.2 Approximation of a Density with Gaussian Mixture
481(3)
11.11 Estimation of Conditional Probability and Conditional Density Functions
484(5)
11.11.1 Estimation of Conditional Probability Functions
484(4)
11.11.2 Estimation of Conditional Density Functions
488(1)
11.12 Connections Between the SV Method and Sparse Function Approximation
489(4)
11.12.1 Reproducing Kernels Hilbert Spaces
490(1)
11.12.2 Modified Sparse Approximation and its Relation to SV Machines
491(2)
12 SV Machines for Pattern Recognition
493(28)
12.1 The Quadratic Optimization Problem
493(3)
12.1.1 Iterative Procedure for Specifying Support Vectors
494(2)
13.1.2 Methods for Solving the Reduced Optimization Problem
496(1)
12.2 Digit Recognition Problem. The U.S. Postal Service Database
496(10)
12.2.1 Performance for the U.S. Postal Service Database
496(4)
12.2.2 Some Important Details
500(3)
12.2.3 Comparison of Performance of the SV Machine with Gaussian Kernel to the Gaussian RBF Network
503(2)
12.2.4 The Best Results for U.S. Postal Service Database
505(1)
12.3 Tangent Distance
506(5)
12.4 Digit Recognition Problem. The NIST Database
511(3)
12.4.1 Performance for NIST Database
511(1)
12.4.2 Further Improvement
512(1)
12.4.3 The Best Results for NIST Database
512(2)
12.5 Furture Racing
514(7)
12.5.1 One More Opportunity. The Transductive Inference
518(3)
13 SV Machines for Function Approximations, Regression Estimation, and Signal Processing
521(50)
13.1 The Model Selection Problem
521(9)
13.1.1 Functional for Model Selection Based on the VC Bound
522(2)
13.1.2 Classical Functionals
524(1)
13.1.3 Experimental Comparison of Model Selection Methods
525(1)
13.1.4 The Problem of Feature Selection Has No General Solution
526(4)
13.2 Structure on the Set of Regularized Linear Functions
530(13)
13.2.1 The L-Curve Method
532(2)
13.2.2 The Method of Effective Number of Parameters
534(2)
13.2.3 The Method of Effective VC Dimension
536(4)
13.2.4 Experiments on Measuring the Effective VC Dimension
540(3)
13.3 Function Approximation Using the SV Method
543(6)
13.3.1 Why Does the Value of Epsilon Control the Number of Support Vectors?
546(3)
13.4 SV Machine for Regression Estimation
549(9)
13.4.1 Problem of Data Smoothing
549(1)
13.4.2 Estimation of Linear Regression Functions
550(6)
13.4.3 Estimation of Nonlinear Regression Functions
556(2)
13.5 SV Method for Solving the Position Emission Tomography (PET) Problem
558(9)
13.5.1 Description of PET
558(2)
13.5.2 Problem of Solving the Radon Equation
560(1)
13.5.3 Generalization of the Residual Principle of Solving PET Problems
561(1)
13.5.4 The Classical Methods of Solving the PET Problem
562(1)
13.5.5 The SV Method for Solving the PET Problem
563(4)
13.6 Remark About the SV Method
567(4)
III STATISTICAL FOUNDATION OF LEARNING THEORY 571(110)
14 Necessary and Sufficient Conditions for Uniform Convergence of Frequencies to Their Probabilities
571(26)
14.1 Uniform Convergence of Frequencies to their Probabilities
572(1)
14.2 Basic Lemma
573(3)
14.3 Entropy of the Set of Events
576(2)
14.4 Asymptotic Properties of the Entropy
578(6)
14.5 Necessary and Sufficient Conditions of Uniform Convergence. Proof of Sufficiency
584(3)
14.6 Necessary and Sufficient Conditions of Uniform Convergence. Proof of Necessity
587(5)
14.7 Necessary and Sufficient Conditions. Continuation of Proving Necessity
592(5)
15 Necessary and Sufficient Conditions for Uniform Convergence of Means to Their Expectations
597(32)
15.1 Epsilon Entropy
597(6)
15.1.1 Proof of the Existence of the Limit
600(1)
15.1.2 Proof of the Convergence of the Sequence
601(2)
15.2 The Quasicube
603(5)
15.3 Epsilon-Extension of a Set
608(2)
15.4 An Auxiliary Lemma
610(4)
15.5 Necessary and Sufficient Conditions for Uniform Convergence. The Proof of Necessity
614(4)
15.6 Necessary and Sufficient Conditions for Uniform Convergence. The Proof of Sufficiency
618(6)
15.7 Corollaries from Theorem 15.1
624(5)
16 Necessary and Sufficient Conditions for Uniform One-Sided Convergence of Means to Their Expectations
629(52)
16.1 Introduction
629(1)
16.2 Maximum Volume Sections
630(6)
16.3 The Theorem on the Average Logarithm
636(6)
16.4 Theorem on the Existence of the Corridor
642(9)
16.5 Theorem on the Existence of Functions Close to the Corridor Boundaries (Theorem on Potential Nonfalsifiability)
651(9)
16.6 The Necessary Conditions
660(6)
16.7 The Necessary and Sufficient Conditions
666(15)
Comments and Bibliographical Remarks 681(42)
References 723(10)
Index 733

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