An introduction to measure-theoretic probability / George G. Roussas.

Format:

Book

Author/Creator:

Roussas, George G.

Language:

English

Subjects (All):

Probabilities.

Measure theory.

Physical Description:

1 online resource (462 p.)

Place of Publication:

Amsterdam : Elsevier Academic Press, c2005.

Language Note:

English

Summary:

This book provides in a concise, yet detailed way, the bulk of the probabilistic tools that a student working toward an advanced degree in statistics,probability and other related areas, should be equipped with. The approach is classical, avoiding the use of mathematical tools not necessary for carrying out the discussions. All proofs are presented in full detail.* Excellent exposition marked by a clear, coherent and logical devleopment of the subject* Easy to understand, detailed discussion of material* Complete proofs

Contents:

Front Cover; An Introduction to Measure-Theoretic Probability; Copyright Page; TABLE OF CONTENTS; PREFACE; CHAPTER 1. Certain Classes of Sets, Measurability, and Pointwise Approximation; 1. Measurable spaces; 2. Product measurable spaces; 3. Measurable functions and random variables; Exercises; CHAPTER 2. Definition and Construction of a Measure and Its Basic Properties; 1. About measures in general, and probability measures in particular; 2. Outer measures; 3. The Carathéodory extension theorem; 4. Measures and (point) functions; Exercises

CHAPTER 3. Some Modes of Convergence of Sequences of Random Variables and Their Relationships1. Almost everywhere convergence and convergence in measure; 2. Convergence in measure is equivalent to mutual convergence in measure; Exercises; CHAPTER 4. The Integral of a Random Variable and Its Basic Properties; 1. Definition of the integral; 2. Basic properties of the integral; 3. Probability distributions; Exercises; CHAPTER 5: Standard Convergence Theorems, the Fubini Theorem; 1. Standard convergence theorems and some of their ramifications

2. Sections, Product Measure Theorem, the Fubini TheoremExercises; CHAPTER 6: Standard Moment and Probability Inequalities, Convergence in the rth Mean and Its Implications; 1. Moment and probability inequalities; 2. Convergence in the rth mean, uniform continuity, uniform integrability, and their relationships; Exercises; CHAPTER 7. The Hahn-Jordan Decomposition Theorem, the Lebesgue Decomposition Theorem, and the Radon-Nikodym Theorem; 1. The Hahn-Jordan Decomposition Theorem; 2. The Lebesgue Decomposition Theorem; 3. The Radon-Nikodym Theorem; Exercises

CHAPTER 8. Distribution Functions and Their Basic Properties, Helly-Bray Type Results1. Basic properties of distribution functions; 2. Weak convergence and compactness of a sequence of distribution functions; 3. Helly-Bray type theorems for distribution functions; Exercises; CHAPTER 9. Conditional Expectation and Conditional Probability, and Related Properties and Results; 1. Definition of conditional expectation and conditional probabilitiy; 2. Some basic theorems about conditional expectations and conditional probabilities

3. Convergence theorems and inequalities for conditional expectations4. Further properties of conditional expectations and conditional probabilities; Exercises; CHAPTER 10. Independence; 1. Independence of events, a-fields, and random variables; 2. Some auxiliary results; 3. Proof of Theorem 1, and of Lemma 1 in Chapter 9; Exercises; CHAPTER 11. Topics from the Theory of Characteristic Functions; 1. Definition of the characteristic function of a distribution and basic properties; 2. The inversion formula

3. Convergence in distribution and convergence of characteristic functions-the Paul Lévy continuity theorem

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

1-280-58200-6

9786613611789

0-08-057530-7

OCLC:

787847128