Galen Shorack, PhD, is Professor Emeritus in the Department of Statistics (of which he was a founding member) and Adjunct Professor in the Department of Mathematics at the University of Washington, Seattle.  He received his Bachelor of Science and Master of Science degrees in Mathematics from the University of Oregon and his PhD in Statistics from Stanford University.  Dr. Shorack's research interests include limit theorems in statistics, the theory of empirical processes, trimming-Winsorizing, and regular variation.  He has served as Associate Editor of the Annals of Mathematical Statistics (Annals of Statistics) and is Fellow of the Institute of Mathematical Statistics.

Preface

Use of This Text

Definition of Symbols

Chapter 1. Measures

1. Basic Properties of Measures
2. Construction and Extension of Measures
3. Lebesgue Stieltjes Measures

1. Mappings and s-Fields
2. Measurable Functions
3. Convergence
4. Probability, RVs, and Convergence in Law
5. Discussion of Sub s-Fields

Chapter 3. Integration

1. The Lebesgue Integral
2. Fundamental Properties of Integrals
3. Evaluating and Differentiating Integrals
4. Inequalities
5. Modes of Convergence

Chapter 4 Derivatives via Signed Measures

1. Introduction
2. Decomposition of Signed Measures
3. The Radon Nikodym Theorem
4. Lebesgue's Theorem
5. The Fundamental Theorem of Calculus

Chapter 5. Measures and Processes on Products

1. Finite-Dimensional Product Spaces
2. Random Vectors on (O,¿,P)
3. Countably Infinite Product Probability Spaces
4. Random Elements and Processes on (O,¿,P)

1. Character of Distribution Functions
2. Properties of Distribution Functions
3. The Quantile Transformation
4. Integration by Parts Applied to Moments
5. Important Statistical Quantities
6. Infinite Variances

Chapter 7. Independence and Conditional Distributions

1. Independence
2. The Tail s-Field
3. Uncorrelated Random Variables
4. Basic Properties of Conditional Expectation
5. Regular Conditional Probability

Chapter 8. WLLN, SLLN, LIL, and Series

1. Introduction
2. Borel Cantelli and Kronecker Lemmas
3. Truncation, WLLN, and Review of Inequalities
4. Maximal Inequalities and Symmetrization
5. The Classical Laws of Large Numbers (or, LLNs)
6. Applications of the Laws of Large Numbers
7. Law of the Iterated Logarithm (or, LIL)
8. Strong Markov Property for Sums of IID RVs
9. Convergence of Series of Independent RVs
10. Martinagles
11. Maximal Inequalities, Some with ¿ Boundaries

1. Classical Convergence in Distribution
2. Determining Classes of Functions
3. Characteristic Functions, with Basic Results
4. Uniqueness and Inversion
5. The Continuity Theorem
6. Elementary Complex and Fourier Analysis
7. Esseen's Lemma
8. Distributions on Grids
9. Conditions for Ø to Be a Characteristic Function

Chapter 10. CLTs via Characteristic Functions

1. Introduction
2. Basic Limit Theorems
3. Variations on the Classical CLT
4. Examples of Limiting Distributions
5. Local Limit Theorems
6. Normality Via Winsorization and Truncation
7. Identically Distributed RVs
8. A Converse of the Classical CLT
9. Bootstrapping
10. Bootstrapping with Slowly ¿ Winsorization

1. Infinitely Divisible Distributions
2. Stable Distributions
3. Characterizing Stable Laws
4. The Domain of Attraction of a Stable Law
5. Gamma Approximations
6. Edgeworth Expansions

Chapter 12. Brownian Motion and Empirical Processes

1. Special Spaces
2. Existence of Processes on (C, C) and (D, D)
3. Brownian Motion and Brownian Bridge
4. Stopping Times
5. Strong Markov Property
6. Embedding a RV in Brownian Motion
7. Barrier Crossing Probabilities
8. Embedding the Partial Sum Process
9. Other Properties of Brownian Motion
10. Various Empirical Processes
11. Inequalities for the Various Empirical Processes
12. Applications

Chapter 13. Martingales

1. Basic Technicalities for Martingales
2. Simple Optional Sampling Theorem
3. The Submartingale Convergence Theorem
4. Applications of the S-mg Convergence Theorem
5. Decomposition of a Submartingale Sequence
6. Optional Sampling
7. Applications of Optional Sampling
8. Introduction to Counting Process Martingales
9. CLTs for Dependent RVs

Chapter 14. Convergence in Law on Metric Spaces

1. Convergence in Distribution on Metric Spaces
2. Metrics for Convergence in Distribution

Chapter 15. Asymptotics Via Empirical Processes

1. Introduction
2. Trimmed and Winsorized Means
3. Linear Rank Statistics and Finite Sampling
4. L-Statistics

Appendix A. Special Distributions

Elementary Probability
1. Distribution Theory for Statistics

Appendix B. General Topology and Hilbert Space

1. General Topology
2. Metric Spaces
3. Hilbert Space
Appendix C. More WLLN and CLT
1. Introduction
2. General Moment Estimation Specific
3. Slowly Varying Partial Variance when s2=8
4. Specific Tail Relationships
5. Regularly Varying Functions
6. Some Winsorized Variance Comparison
7. Inequalities for Winsorized Quantile Functions

References

Index

The choice of examples used in this text clearly illustrate its use for a one-year graduate course. The material to be presented in the classroom constitutes a little more than half the text, while the rest of the text provides background, offers different routes that could be pursued in the classroom, as well as additional material that is appropriate for self-study. Of particular interest is a presentation of the major central limit theorems via Steins method either prior to or alternative to a characteristic function presentation. Additionally, there is considerable emphasis placed on the quantile function as well as the distribution function, with both the bootstrap and trimming presented. The section on martingales covers censored data martingales.