# Uniform Central Limit Theorems

@inproceedings{Dudley1999UniformCL, title={Uniform Central Limit Theorems}, author={Richard M. Dudley}, year={1999} }

Preface 1. Introduction: Donsker's theorem, metric entropy and inequalities 2. Gaussian measures and processes sample continuity 3. Foundations of uniform central limit theorems: Donsker classes 4. Vapnik-Cervonenkis combinatorics 5. Measurability 6. Limit theorems for Vapnik-Cervonenkis and related classes 7. Metric entropy, with inclusion and bracketing 8. Approximation of functions and sets 9. Sums in general Banach spaces and invariance principles 10. Universal and uniform central limit…

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## References

SHOWING 1-10 OF 201 REFERENCES

### Real Analysis and Probability

- Mathematics
- 1989

1. Foundations: set theory 2. General topology 3. Measures 4. Integration 5. Lp spaces: introduction to functional analysis 6. Convex sets and duality of normed spaces 7. Measure, topology, and…

### The central limit theorem for non-separable valued functions

- Mathematics
- 1985

The main purpose of this paper is to formulate and investigate the central limit theorem for functions which are not assumed to be separable-valued nor measurable. The inspiration is a part of a…

### Real and complex analysis

- Mathematics
- 1966

Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures…

### Weak Convergence and Empirical Processes: With Applications to Statistics

- Mathematics
- 1996

This chapter discusses Convergence: Weak, Almost Uniform, and in Probability, which focuses on the part of Convergence of the Donsker Property which is concerned with Uniformity and Metrization.

### On Hoffmann-Jørgensen-type Inequalities for Outer Expectations with Applications

- Mathematics
- 1997

The paper unifies and extends a number of Hoffmann-j0rgensen-type inequalities known in the literature. Originally proved for sums of independent Banach-space valued random variables and commonly…

### A Maximal Inequality and a Functional Central Limit Theorem for set-indexed empirical processes

- Mathematics
- 1997

For the tail probabilities of a general set-indexed empirical process in an arbitrary sample space a maximal inequality is derived. In the case that the class of sets by which the process is indexed…

### Probability Inequalities for Empirical Processes and a Law of the Iterated Logarithm

- Mathematics
- 1984

Given a class -F of functions on X, we can view Vn as a stochastic process indexed by -F and consider limit theorems for this process. To prove such theorems it is often helpful to have bounds on the…

### Convergence of stochastic processes

- Mathematics
- 1984

I Functional on Stochastic Processes.- 1. Stochastic Processes as Random Functions.- Notes.- Problems.- II Uniform Convergence of Empirical Measures.- 1. Uniformity and Consistency.- 2. Direct…

### Weak Convergence of Probabilities on Nonseparable Metric Spaces and Empirical Measures on Euclidean Spaces

- Mathematics
- 1966

It is known that under certain mild set-theoretic assumptions, a finite, countably additive measure defined on all Borel sets of a metric space is concentrated in a separable subspace (Marczewski and…