Probability for Statisticians - Softcover

Shorack, Galen R.

Softcover

ISBN 10: 3319522086 ISBN 13: 9783319522081

Publisher: Springer, 2017

Synopsis

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

Chapter 2. Measurable Functions and Convergence

1. Mappings and σ-Fields
2. Measurable Functions
3. Convergence
4. Probability, RVs, and Convergence in Law
5. Discussion of Sub σ-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 (Ω,Α,P)
3. Countably Infinite Product Probability Spaces
4. Random Elements and Processes on (Ω,Α,P)

Chapter 6. Distribution and Quantile Functions

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 σ-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

Chapter 9. Characteristic Functions and Determining Classes

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

Chapter 11. Infinitely Divisible and Stable Distributions

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. Var

Edition details

• Publisher: Springer
• Publication date: 2017
• Language: English
• ISBN 10: 3319522086
• ISBN 13: 9783319522081
• Binding: Paperback