CART

(0) items

Probability and Measure,9781118122372
This item qualifies for
FREE SHIPPING!
FREE SHIPPING OVER $59!

Your order must be $59 or more, you must select US Postal Service Shipping as your shipping preference, and the "Group my items into as few shipments as possible" option when you place your order.

Bulk sales, PO's, Marketplace Items, eBooks, Apparel, and DVDs not included.

Probability and Measure

by
Edition:
4th
ISBN13:

9781118122372

ISBN10:
1118122372
Format:
Hardcover
Pub. Date:
2/28/2012
Publisher(s):
Wiley

Questions About This Book?

Why should I rent this book?
Renting is easy, fast, and cheap! Renting from eCampus.com can save you hundreds of dollars compared to the cost of new or used books each semester. At the end of the semester, simply ship the book back to us with a free UPS shipping label! No need to worry about selling it back.
How do rental returns work?
Returning books is as easy as possible. As your rental due date approaches, we will email you several courtesy reminders. When you are ready to return, you can print a free UPS shipping label from our website at any time. Then, just return the book to your UPS driver or any staffed UPS location. You can even use the same box we shipped it in!
What version or edition is this?
This is the 4th edition with a publication date of 2/28/2012.
What is included with this book?
  • The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any CDs, lab manuals, study guides, etc.
  • The Rental copy of this book is not guaranteed to include any supplemental materials. You may receive a brand new copy, but typically, only the book itself.

Summary

Now re-issued in a new style and format, but with the same reliable content of the Third Edition, Probability and Measure offers advanced students, scientists, and engineers an integrated introduction to measure theory and probability. Retaining the unique approach of the previous editions, this text interweaves material on probability and measure so that probability problems generate an interest in measure theory and measure theory is, then, developed and applied to probability. Probability and Measure provides a thorough coverage of probability, measure, integration, random variables and expected values, convergence of distributions, derivatives and conditional probability, and stochastic processes. In recognition of over thirty years of publication and having impacted tens of thousands of readers, the 2012 Edition features an open and user-friendly design, updated bibliography, improved treatment of Brownian motion, and replacement of queuing theory with ergodic theory. Like its previous editions, this re-issue will be well received by students of mathematics, statistics, economics, and a wide variety of disciplines that require a solid understanding of probability theory. An extensive supplement of additional notes on the problems and chapter commentaries is available at further cost either independently or as a set with the book.

Author Biography

Patrick Billingsley was Professor Emeritus of Statistics and Mathematics at the University of Chicago and a world-renowned authority on probability theory before his untimely death in 2011. He was the author of Convergence of Probability Measures (Wiley), among other works. Dr. Billingsley edited the Annals of Probability for the Institute of Mathematical Statistics. He received his PhD in mathematics from Princeton University.

Table of Contents

FOREWORD xi

PREFACE xiii

Patrick Billingsley 1925–2011 xv

Chapter1 PROBABILITY 1

1. BOREL’S NORMAL NUMBER THEOREM, 1

The Unit Interval

The Weak Law of Large Numbers

The Strong Law of Large Numbers

Strong Law Versus Weak

Length

The Measure Theory of Diophantine Approximation*

2. PROBABILITY MEASURES, 18

Spaces

Assigning Probabilities

Classes of Sets

Probability Measures

Lebesgue Measure on the Unit Interval

Sequence Space*

Constructing s-Fields*

3. EXISTENCE AND EXTENSION, 39

Construction of the Extension

Uniqueness and the p? Theorem

Monotone Classes

Lebesgue Measure on the Unit Interval

Completeness

Nonmeasurable Sets

Two Impossibility Theorems*

4. DENUMERABLE PROBABILITIES, 53

General Formulas

Limit Sets

Independent Events

Subfields

The Borel-Cantelli Lemmas

The Zero-One Law

5. SIMPLE RANDOM VARIABLES, 72

Definition

Convergence of Random Variables

Independence

Existence of Independent Sequences

Expected Value

Inequalities

6. THE LAW OF LARGE NUMBERS, 90

The Strong Law

The Weak Law

Bernstein's Theorem

A Refinement of the Second Borel-Cantelli Lemma

7. GAMBLING SYSTEMS, 98

Gambler's Ruin

Selection Systems

Gambling Policies

Bold Play*

Timid Play*

8. MARKOV CHAINS, 117

Definitions

Higher-Order Transitions

An Existence Theorem

Transience and Persistence

Another Criterion for Persistence

Stationary Distributions

Exponential Convergence*

Optimal Stopping*

9. LARGE DEVIATIONS AND THE LAW OF THE ITERATED LOGARITHM, 154

Moment Generating Functions

Large Deviations

Chernoff's Theorem*

The Law of the Iterated Logarithm

Chapter2 MEASURE 167

10. GENERAL MEASURES, 167

Classes of Sets

Conventions Involving 8

Measures

Uniqueness

11. OUTER MEASURE, 174

Outer Measure

Extension

An Approximation Theorem

12. MEASURES IN EUCLIDEAN SPACE, 181

Lebesgue Measure

Regularity

Specifying Measures on the Line

Specifying Measures in Rk

Strange Euclidean Sets*

13. MEASURABLE FUNCTIONS AND MAPPINGS, 192

Measurable Mappings

Mappings into Rk

Limits and Measurability

Transformations of Measures

14. DISTRIBUTION FUNCTIONS, 198

Distribution Functions

Exponential Distributions

Weak Convergence

Convergence of Types*

Extremal Distributions*

Chapter3 INTEGRATION 211

15. THE INTEGRAL, 211

Definition

Nonnegative Functions

Uniqueness

16. PROPERTIES OF THE INTEGRAL, 218

Equalities and Inequalities

Integration to the Limit

Integration over Sets

Densities

Change of Variable

Uniform Integrability

Complex Functions

17. THE INTEGRAL WITH RESPECT TO LEBESGUE MEASURE, 234

The Lebesgue Integral on the Line

The Riemann Integral

The Fundamental Theorem of Calculus

Change of Variable

The Lebesgue Integral in Rk

Stieltjes Integrals

18. PRODUCT MEASURE AND FUBINI’S THEOREM, 245

Product Spaces

Product Measure

Fubini's Theorem

Integration by Parts

Products of Higher Order

19. THE Lp SPACES*, 256

Definitions

Completeness and Separability

Conjugate Spaces

Weak Compactness

Some Decision Theory

The Space L2

An Estimation Problem

Chapter4 RANDOM VARIABLES AND EXPECTED VALUES 271

20. RANDOM VARIABLES AND DISTRIBUTIONS, 271

Random Variables and Vectors

Subfields

Distributions

Multidimensional Distributions

Independence

Sequences of Random Variables

Convolution

Convergence in Probability

The Glivenko-Cantelli Theorem*

21. EXPECTED VALUES, 291

Expected Value as Integral

Expected Values and Limits

Expected Values and Distributions

Moments

Inequalities

Joint Integrals

Independence and Expected Value

Moment Generating Functions

22. SUMS OF INDEPENDENT RANDOM VARIABLES, 300

The Strong Law of Large Numbers

The Weak Law and Moment Generating Functions

Kolmogorov's Zero-One Law

Maximal Inequalities

Convergence of Random Series

Random Taylor Series*

23. THE POISSON PROCESS, 316

Characterization of the Exponential Distribution

The Poisson Process

The Poisson Approximation

Other Characterizations of the Poisson Process

Stochastic Processes

24. THE ERGODIC THEOREM*, 330

Measure-Preserving Transformations

Ergodicity

Ergodicity of Rotations

Proof of the Ergodic Theorem

The Continued-Fraction Transformation

Diophantine Approximation

Chapter5 CONVERGENCE OF DISTRIBUTIONS 349

25. WEAK CONVERGENCE, 349

Definitions

Uniform Distribution Modulo 1*

Convergence in Distribution

Convergence in Probability

Fundamental Theorems

Helly's Theorem

Integration to the Limit

26. CHARACTERISTIC FUNCTIONS, 365

Definition

Moments and Derivatives

Independence

Inversion and the Uniqueness Theorem

The Continuity Theorem

Fourier Series*

27. THE CENTRAL LIMIT THEOREM, 380

Identically Distributed Summands

The Lindeberg and Lyapounov Theorems

Dependent Variables*

28. INFINITELY DIVISIBLE DISTRIBUTIONS*, 394

Vague Convergence

The Possible Limits

Characterizing the Limit

29. LIMIT THEOREMS IN Rk, 402

The Basic Theorems

Characteristic Functions

Normal Distributions in Rk

The Central Limit Theorem

30. THE METHOD OF MOMENTS*, 412

The Moment Problem

Moment Generating Functions

Central Limit Theorem by Moments

Application to Sampling Theory

Application to Number Theory

Chapter6 DERIVATIVES AND CONDITIONAL PROBABILITY 425

31. DERIVATIVES ON THE LINE*, 425

The Fundamental Theorem of Calculus

Derivatives of Integrals

Singular Functions

Integrals of Derivatives

Functions of Bounded Variation

32. THE RADON–NIKODYM THEOREM, 446

Additive Set Functions

The Hahn Decomposition

Absolute Continuity and Singularity

The Main Theorem

33. CONDITIONAL PROBABILITY, 454

The Discrete Case

The General Case

Properties of Conditional Probability

Difficulties and Curiosities

Conditional Probability Distributions

34. CONDITIONAL EXPECTATION, 472

Definition

Properties of Conditional Expectation

Conditional Distributions and Expectations

Sufficient Subfields*

Minimum-Variance Estimation*

35. MARTINGALES, 487

Definition

Submartingales

Gambling

Functions of Martingales

Stopping Times

Inequalities

Convergence Theorems

Applications: Derivatives

Likelihood Ratios

Reversed Martingales

Applications: de Finetti's Theorem

Bayes Estimation

A Central Limit Theorem*

Chapter7 STOCHASTIC PROCESSES 513

36. KOLMOGOROV'S EXISTENCE THEOREM, 513

Stochastic Processes

Finite-Dimensional Distributions

Product Spaces

Kolmogorov's Existence Theorem

The Inadequacy of RT

A Return to Ergodic Theory

The HewittSavage Theorem*

37. BROWNIAN MOTION, 530

Definition

Continuity of Paths

Measurable Processes

Irregularity of Brownian Motion Paths

The Strong Markov Property

The Reflection Principle

Skorohod Embedding

Invariance*

38. NONDENUMERABLE PROBABILITIES, 558

Introduction

Definitions

Existence Theorems

Consequences of Separability*

APPENDIX 571

NOTES ON THE PROBLEMS 587

BIBLIOGRAPHY 617

INDEX 619



Please wait while the item is added to your cart...