Strong Approximations in Probability and Statistics

Preface

Introduction


1. Wiener and Some Related Gaussian Processes


1.0 On the Notion of a Wiener Process


1.1 Definition and Existence of a Wiener Process


1.2 How Big are the Increments of a Wiener Process?


1.3 The Law of Iterated Logarithm for the Wiener Process


1.4 Brownian Bridges


1.5 The Distributions of Some Functional of the Wiener and Brownian Bridge Processes


1.6 The Modulus of Non-Differentiability of the Wiener Process


1.7 How Small are the Increments of a Wiener Process?


1.8 Infinite Series Representations of the Wiener Process and Brownian Bridge


1.9 The Ornstein-Uhlenbeck Process


1.10 On the Notion of a Two-Parameter Wiener Process


1.11 Definition and Existence of a Two-Parameter Wiener Process


1.12 How Big are the Increments of a Two-Parameter Wiener Process?


1.13 A Continuity Modulus of W(x, y)


1.14 The Limit Points of W(x, y) as y → ∞


1.15 The Kiefer Process

Supplementary Remarks


2. Strong Approximations of Partial Sums of I.I.D.R.V. by Wiener Processes


2.0 Notations


2.1 A Proof of Donsker's Theorem with Skorohod's Embedding Scheme


2.2 The Strong Invariance Principle Appears


2.3 The Stochastic Geyser Problem as a Lower Limit to the Strong Invariance Problem


2.4 The Longest Runs of Pure Heads and the Stochastic Geyser Problem


2.5 Improving the Upper Limit


2.6 The Best Rates Emerge

Supplementary Remarks


3. A Study of Partial Sums with the Help of Strong Approximation Methods


3.0 Introduction


3.1 How Big are the Increments of Partial Sums of I.I.D.R.V. when the Moment Generating Function Exists?


3.2 How Big are the Increments of Partial Sums of I.I.D.R.V. when the Moment Generating Function Does not Exist?


3.3 How Small are the Increments of Partial Sums of I.I.D.R.V.?


3.4 A Summary

Supplementary Remarks


4. Strong Approximations of Empirical Processes by Gaussian Processes


4.1 Some Classical Results


4.2 Why Should the Empirical Process Behave Like a Brownian Bridge?


4.3 The First Strong Approximations of the Empirical Process


4.4 Best Strong Approximations of the Empirical Process


4.5 Strong Approximation of the Quantile Process

Supplementary Remarks


5. A Study of Empirical and Quantile Processes with the Help of Strong Approximation Methods


5.0 Introduction


5.1 The Law of Iterated Logarithm for the Empirical Process


5.2 The Distance Between the Empirical and the Quantile Processes


5.3 The Law of Iterated Logarithm for the Quantile Process


5.4 Asymptotic Distribution Results for Some Classical Functionals of the Empirical Process


5.5 Asymptotic Distribution Results for Some Classical Functionals of the Quantile Process


5.6 Asymptotic Distribution Results for Some Classical Functionals of Some k-Sample Empirical and Quantile Processes


5.7 Approximations of the Empirical Process when Parameters are Estimated


5.8 Asymptotic Quadratic Quantile Tests for Composite Goodness-of-Fit


5.9 On Testing for Exponentiality

Supplementary Remarks


6. A Study of Further Empirical Processes with the Help of Strong Approximation Methods


6.0 Introduction


6.1 Strong Invariance Principles and Limit Distributions for Empirical Densities


6.2 Strong Theorems for Empirical Densities


6.3 Empirical Regression


6.4 Empirical Characteristic Functions

Supplementary Remarks


7. Random Limit Theorems Via Strong Invariance Principles


7.0 Introduction and Some Historical Remarks


7.1 Laws of Large Numbers for Randomly Selected Sequences


7.2 Invariance (Strong and Weak) Principles for Random-Sum Limit Theorems


7.3 Invariance (Strong and Weak) Principles for Random Size Empirical Processes

References

Author Index

Subject Index

Summary of Notations and Abbreviations