Probability and Measure Theory

ISBN-10: 0120652021

ISBN-13: 9780120652020

Edition: 2nd 2000 (Revised)

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Description: Intended for a graduate course in probability that includes background topics in analysis, this text includes coverage of conditional probability and expectation, strong laws or large numbers, martingale theory, and the central limit theorem.

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Book details

List price: $129.00
Edition: 2nd
Copyright year: 2000
Publisher: Elsevier Science & Technology Books
Publication date: 12/6/1999
Binding: Hardcover
Pages: 516
Size: 6.50" wide x 9.50" long x 1.50" tall
Weight: 2.354
Language: English

Preface
Summary of Notation
Fundamentals of Measure and Integration Theory
Introduction
Fields, [sigma]-Fields, and Measures
Extension of Measures
Lebesgue-Stieltjes Measures and Distribution Functions
Measurable Functions and Integration
Basic Integration Theorems
Comparison of Lebesgue and Riemann Integrals
Further Results in Measure and Integration Theory
Introduction
Radon-Nikodym Theorem and Related Results
Applications to Real Analysis
L[superscript p] Spaces
Convergence of Sequences of Measurable Functions
Product Measures and Fubini's Theorem
Measures on Infinite Product Spaces
Weak Convergence of Measures
References
Introduction to Functional Analysis
Introduction
Basic Properties of Hilbert Spaces
Linear Operators on Normed Linear Spaces
Basic Theorems of Functional Analysis
References
Basic Concepts of Probability
Introduction
Discrete Probability Spaces
Independence
Bernoulli Trials
Conditional Probability
Random Variables
Random Vectors
Independent Random Variables
Some Examples from Basic Probability
Expectation
Infinite Sequences of Random Variables
References
Conditional Probability and Expectation
Introduction
Applications
The General Concept of Conditional Probability and Expectation
Conditional Expectation Given a [sigma]-Field
Properties of Conditional Expectation
Regular Conditional Probabilities
Strong Laws of Large Numbers and Martingale Theory
Introduction
Convergence Theorems
Martingales
Martingale Convergence Theorems
Uniform Integrability
Uniform Integrability and Martingale Theory
Optional Sampling Theorems
Applications of Martingale Theory
Applications to Markov Chains
References
The Central Limit Theorem
Introduction
The Fundamental Weak Compactness Theorem
Convergence to a Normal Distribution
Stable Distributions
Infinitely Divisible Distributions
Uniform Convergence in the Central Limit Theorem
The Skorokhod Construction and Other Convergence Theorems
The k-Dimensional Central Limit Theorem
References
Ergodic Theory
Introduction
Ergodicity and Mixing
The Pointwise Ergodic Theorem
Applications to Markov Chains
The Shannon-McMillan Theorem
Entropy of a Transformation
Bernoulli Shifts
References
Brownian Motion and Stochastic Integrals
Stochastic Processes
Brownian Motion
Nowhere Differentiability and Quadratic Variation of Paths
Law of the Iterated Logarithm
The Markov Property
Martingales
Ito Integrals
Ito's Differentiation Formula
References
Appendices
The Symmetric Random Walk in R[superscript k]
Semicontinuous Functions
Completion of the Proof of Theorem 7.3.2
Proof of the Convergence of Types Theorem 7.3.4
The Multivariate Normal Distribution
Bibliography
Solutions to Problems
Index
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