Probability and Measure Theory

by Ash; Doleans-Dade

• ISBN13: 9780120652020
• ISBN10: 0120652021
• Edition: 2nd
• Format: Hardcover
• Copyright: 1999-12-06
• Publisher: Elsevier Science

Summary

Probability and Measure Theory, Second Edition, is a text for a graduate-level course in probability that includes essential background topics in analysis. It provides extensive coverage of conditional probability and expectation, strong laws of large numbers, martingale theory, the central limit theorem, ergodic theory, and Brownian motion. * Clear, readable style * Solutions to many problems presented in text * Solutions manual for instructors * Material new to the second edition on ergodic theory, Brownian motion, and convergence theorems used in statistics * No knowledge of general topology required, just basic analysis and metric spaces * Efficient organization

Table of Contents

Preface	vii
Summary of Notation	ix
Fundamentals of Measure and Integration Theory	1	(59)
Introduction	1	(2)
Fields, σ-Fields, and Measures	3	(9)
Extension of Measures	12	(10)
Lebesgue-Stieltjes Measures and Distribution Functions	22	(13)
Measurable Functions and Integration	35	(10)
Basic Integration Theorems	45	(10)
Comparison of Lebesgue and Riemann Integrals	55	(5)
Further Results in Measure and Integration Theory	60	(67)
Introduction	60	(4)
Radon-Nikodym Theorem and Related Results	64	(8)
Applications to Real Analysis	72	(11)
Lp Spaces	83	(13)
Convergence of Sequences of Measurable Functions	96	(5)
Product Measures and Fubini's Theorem	101	(12)
Measures on Infinite Product Spaces	113	(8)
Weak Convergence of Measures	121	(4)
References	125	(2)
Introduction to Functional Analysis	127	(39)
Introduction	127	(3)
Basic Properties of Hilbert Spaces	130	(11)
Linear Operators on Normed Linear Spaces	141	(11)
Basic Theorems of Functional Analysis	152	(13)
References	165	(1)
Basic Concepts of Probability	166	(35)
Introduction	166	(1)
Discrete Probability Spaces	167	(1)
Independence	167	(3)
Bernoulli Trials	170	(1)
Conditional Probability	171	(2)
Random Variables	173	(3)
Random Vectors	176	(2)
Independent Random Variables	178	(3)
Some Examples from Basic Probability	181	(7)
Expectation	188	(8)
Infinite Sequences of Random Variables	196	(4)
References	200	(1)
Conditional Probability and Expectation	201	(34)
Introduction	201	(1)
Applications	202	(2)
The General Concept of Conditional Probability and Expectation	204	(11)
Conditional Expectation Given a o-Field	215	(5)
Properties of Conditional Expectation	220	(8)
Regular Conditional Probabilities	228	(7)
Strong Laws of Large Numbers and Martingale Theory	235	(55)
Introduction	235	(4)
Convergence Theorems	239	(9)
Martingales	248	(9)
Martingale Convergence Theorems	257	(5)
Uniform Integrability	262	(4)
Uniform Integrability and Martingale Theory	266	(4)
Optional Sampling Theorems	270	(7)
Applications of Martingale Theory	277	(8)
Applications to Markov Chains	285	(3)
References	288	(2)
The Central Limit Theorem	290	(55)
Introduction	290	(10)
The Fundamental Weak Compactness Theorem	300	(7)
Convergence to a Normal Distribution	307	(10)
Stable Distributions	317	(3)
Infinitely Divisible Distributions	320	(9)
Uniform Convergence in the Central Limit Theorem	329	(3)
The Skorokhod Construction and Other Convergence Theorems	332	(4)
The k-Dimensional Central Limit Theorem	336	(8)
References	344	(1)
Ergodic Theory	345	(54)
Introduction	345	(5)
Ergodicity and Mixing	350	(6)
The Pointwise Ergodic Theorem	356	(12)
Applications to Markov Chains	368	(6)
The Shannon-McMillan Theorem	374	(12)
Entropy of a Transformation	386	(8)
Bernoulli Shifts	394	(3)
References	397	(2)
Brownian Motion and Stochastic Integrals	399	(39)
Stochastic Processes	399	(2)
Brownian Motion	401	(7)
Nowhere Differentiability and Quadratic Variation of Paths	408	(2)
Law of the Iterated Logarithm	410	(4)
The Markov Property	414	(6)
Martingales	420	(6)
Ito Integrals	426	(6)
Ito's Differentiation Formula	432	(5)
References	437	(1)
Appendices	438	(16)
1. The Symmetric Random Walk in Rk	438	(3)
2. Semicontinuous Functions	441	(2)
3. Completion of the Proof of Theorem 7.3.2	443	(4)
4. Proof of the Convergence of Types Theorem 7.3.4	447	(2)
5. The Multivariate Normal Distribution	449	(5)
Bibliography	454	(2)
Solutions to Problems	456	(56)
Index	512