₹ 1625
American Mathematical Society
2012
240
9780821891810
Paperback
English
180 x 240 mm
India,Nepal,Bhutan,Bangladesh,Sri Lanka,Maldives,Pakistan
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.
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
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