Probability
Davar Khoshnevisan
Price
1625
ISBN
9780821891810
Language
English
Pages
240
Format
Paperback
Dimensions
180 x 240 mm
Year of Publishing
2012
Territorial Rights
Restricted
Imprint
American Mathematical Society
Catalogues

This is a textbook for a one-semester graduate course in measure-theoretic probability theory, but with ample material to cover an ordinary year-long course at a more leisurely pace. Khoshnevisan''s approach is to develop the ideas that are absolutely central to modern probability theory, and to showcase them by presenting their various applications. As a result, a few of the familiar topics are replaced by interesting non-standard ones. The topics range from undergraduate probability and classical limit theorems to Brownian motion and elements of stochastic calculus. Throughout, the reader will find many exciting applications of probability theory and probabilistic reasoning. There are numerous exercises, ranging from the routine to the very difficult. Each chapter concludes with historical notes.

Davar Khoshnevisan, University of Utah, Salt Lake City, UT

Preface
General Notation

Chapter 1.   Classical Probability
    1. Discrete Probability
    2. Conditional Probability
    3. Independence
    4. Discrete Distributions
    5. Absolutely Continuous Distributions
    6. ExpectationandVariance
    Problems
    Notes

Chapter 2.  Bernoulli Trials  
    1. The Classical Theorems
    Problems
    Notes

Chapter3.  MeasureTheory
    1. MeasureSpaces
    2. LebesgueMeasure
    3. Completion
    4. Proof of Caratheodory's Theorem
    Problemsbr
    Notes

Chapter 4.  Integration
    1. Measurable Functions
    2. The Abstract Integral
    3. Lp-Spaces
    4. ModesofConvergence
    5. LimitTheorems
    6. The Radon-Nikodym Theorem
    Problems
    Notes

Chapter5.  ProductSpaces
    1. FiniteProducts  
    2. Infinite Products
    3. Complement: Proof of Kolmogorov's Extension Theorem
    Problems
    Notes

Chapter6.  Independence
    1. Random Variables and Distributions
    2. Independent Random Variables
    3. AnInstructiveExample
    4. Khintchine's Weak Law of Large Numbers
    5. Kolmogorov's Strong Law of Large Numbers
    6. Applications
    Problems
    Notes

Chapter7.  TheCentralLimitTheorem
    1. WeakConvergence
    2. Weak Convergence and Compact-Support Functions
    3. Harmonic Analysis in Dimension One
    4. ThePlancherelTheorem
    5. The1-DCentralLimitTheorem
    6. ComplementstotheCLT
    Problems
    Notes

Chapter 8.   Martingales
    1.   Conditional Expectations
    2.   Filtrations and Semi-Martingales
    3.   Stopping Times and Optional Stopping
    4.   Applications to Random Walks
    5.   Inequalities and Convergence
    6.   Further Applications
    Problems
    Notes

Chapter9.   BrownianMotion
    1.   Gaussian Processes
    2.   Wiener's Construction: Brownian Motion on [0, 1)
    3.   Nowhere-Differentiability
    4.   The Brownian Filtration and Stopping Times
    5.   TheStrongMarkovProperty
    6.   The Reflection Principle
    Problems
    Notes

Chapter 10.   Terminus: Stochastic Integration
    1.   The Indefinite Ito Integral
    2.   Continuous Martingales in L2(P)
    3.   The Definite Ito Integral
    4.   Quadratic Variation
    5.   Ito's Formula and Two Applications
    Problems
    Notes

Appendix
    1.   Hilbert Spaces
    2.   FourierSeries
Bibliography
Index