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Davar Khoshnevisan
Price : ₹ 1020.00  
ISBN : 9780821891810
Language : English
Pages : 240
Binding : Paperback
Book Size : 180 x 240 mm
Year : 2012
Series :
Territorial Rights : Restricted
Imprint : American Mathematical Society
Catalogues : EducationMathematics
About the Book

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.

Table of Contents

General Notation

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

Chapter 2.  Bernoulli Trials  
    1. The Classical Theorems

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

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

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

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

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

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

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

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

    1.   Hilbert Spaces
    2.   FourierSeries

Contributors (Author(s), Editor(s), Translator(s), Illustrator(s) etc.)

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