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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.

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Preface Use of This Text Definition of Symbols
Chapter 1. Measures
Basic Properties of Measures
Construction and Extension of Measures
Lebesgue Stieltjes Measures Chapter 2. Measurable Functions and Convergence
Mappings and s-Fields
Measurable Functions
Convergence
Probability, RVs, and Convergence in Law
Discussion of Sub s-Fields Chapter 3. Integration
The Lebesgue Integral
Fundamental Properties of Integrals
Evaluating and Differentiating Integrals
Inequalities
Modes of Convergence Chapter 4 Derivatives via Signed Measures
Introduction
Decomposition of Signed Measures
The Radon Nikodym Theorem
Lebesgue´s Theorem
The Fundamental Theorem of Calculus Chapter 5. Measures and Processes on Products
Finite-Dimensional Product Spaces
Random Vectors on (, ,P )
Countably Infinite Product Probability Spaces
Random Elements and Processes on (, ,P ) Chapter 6. Distribution and Quantile Functions
Character of Distribution Functions
Properties of Distribution Functions
The Quantile Transformation
Integration by Parts Applied to Moments
Important Statistical Quantities
Infinite Variances Chapter 7. Independence and Conditional Distributions
Independence
The Tail s-Field
Uncorrelated Random Variables
Basic Properties of Conditional Expectation
Regular Conditional Probability Chapter 8. WLLN, SLLN, LIL, and Series
Introduction
Borel Cantelli and Kronecker Lemmas
Truncation, WLLN, and Review of Inequalities
Maximal Inequalities and Symmetrization
The Classical Laws of Large Numbers (or, LLNs)
Applications of the Laws of Large Numbers
Law of the Iterated Logarithm (or, LIL)
Strong Markov Property for Sums of IID RVs
Convergence of Series of Independent RVs
Martinagles
Maximal Inequalities, Some with Boundaries Chapter 9. Characteristic Functions and Determining Classes
Classical Convergence in Distribution
Determining Classes of Functions
Characteristic Functions, with Basic Results
Uniqueness and Inversion
The Continuity Theorem
Elementary Complex and Fourier Analysis
Esseen´s Lemma
Distributions on Grids
Conditions for � to Be a Characteristic Function Chapter 10. CLTs via Characteristic Functions
Introduction
Basic Limit Theorems
Variations on the Classical CLT
Examples of Limiting Distributions
Local Limit Theorems
Normality Via Winsorization and Truncation
Identically Distributed RVs
A Converse of the Classical CLT
Bootstrapping
Bootstrapping with Slowly Winsorization Chapter 11. Infinitely Divisible and Stable Distributions
Infinitely Divisible Distributions
Stable Distributions
Characterizing Stable Laws
The Domain of Attraction of a Stable Law
Gamma Approximations
Edgeworth Expansions Chapter 12. Brownian Motion and Empirical Processes
Special Spaces
Existence of Processes on ( C, C ) and ( D, D )
Brownian Motion and Brownian Bridge
Stopping Times
Strong Markov Property
Embedding a RV in Brownian Motion
Barrier Crossing Probabilities
Embedding the Partial Sum Process
Other Properties of Brownian Motion
Various Empirical Processes
Inequalities for the Various Empirical Processes
Applications Chapter 13. Martingales
Basic Technicalities for Martingales
Simple Optional Sampling Theorem
The Submartingale Convergence Theorem
Applications of the S-mg Convergence Theorem
Decomposition of a Submartingale Sequence
Optional Sampling
Applications of Optional Sampling
Introduction to Counting Process Martingales
CLTs for Dependent RVs Chapter 14. Convergence in Law on Metric Spaces
Convergence in Distribution on Metric Spaces
Metrics for Convergence in Distribution Chapter 15. Asymptotics Via Empirical Processes
Introduction
Trimmed and Winsorized Means
Linear Rank Statistics and Finite Sampling
L -Statistics Appendix A. Special Distributions Elementary Probability
Distribution Theory for Statistics Appendix B. General Topology and Hilbert Space
General Topology
Metric Spaces
Hilbert Space Appendix C. More WLLN and CLT
Introd

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