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Probability (Graduate Studies in Mathematics)
By Davar Khoshnevisan
* Publisher: American Mathematical Society
* Number Of Pages: 224
* Publication Date: 2007-03-20
* ISBN-10 / ASIN: 0821842153
* ISBN-13 / EAN: 9780821842157
Product Description:
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.
Contents
Preface
General Notation
Chapter 1. Classical Probability 1
1. Discrete Probability 1
2. Conditional Probability 4
3. Independence 6
4. Discrete Distributions 6
5. Absolutely Continuous Distributions 10
6. Expectation and Variance 12
Problems 13
Notes 15
Chapter 2. Bernoulli Trials 17
1. The Classical Theorems 18
Problems 21
Notes 22
Chapter 3. Measure Theory 23
1. Measure Spaces 23
2. Lebesgue Measure 25
3. Completion 28
4. Proof of Caratheodory's Theorem 30
Problems 33
Notes 34
Chapter 4. Integration 35
1. Measurable Functions 35
2. The Abstract Integral 37
3. LP-Spaces 39
4. Modes of Convergence 4: 3
5. Limit Theorems 45
6. The Radon-Nikodym Theorem 47
Problems 49
Notes 52
Chapter 5. Product Spaces 53
1. Finite Products 53
2. Infinite Products 58
3. Complement: Proof of Kolmogorov's Extension Theorem 60
Problems 62
Notes 64
Chapter 6. Independence 65
1. Random Variables and Distributions 65
2. Independent Random Variables 67
3. An Instructive Example 71
4. Khintchine's Weak Law of Large Numbers 71
5. Kolmogorov's Strong Law of Large Numbers 73
6. Applications 77
Problems 84
Notes 89
Chapter 7. The Central Limit Theorem 91
1. Weak Convergence 91
2. Weak Convergence and Compact-Support Functions 94
3. Harmonic Analysis in Dimension One 96
4. The Plancherel Theorem 97
5. The 1-D Central Limit Theorem 100
6. Complements to the CLT 101
Problems 111
Notes 117
Chapter 8. Martingales 119
1. Conditional Expectations 119
2. Filtrations and Semi-Martingales 126
3. Stopping Times and Optional Stopping 129
4. Applications to Random Walks 131
5. Inequalities and Convergence 134
6. Further Applications 136
Problems 151
Notes 157
Chapter 9. Brownian Motion 159
1. Gaussian Processes 160
2. Wiener's Construction: Brownian Motion on [0.1) 165
3. Nowhere-Differentiability 168
4. The Brownian Filtration and Stopping Times 170
5. The Strong Markov Property 17. 3
6. The Reflection Principle 175
Problems 176
Notes 180
Chapter 10. Terminus: Stochastic Integration 181
1. The Indefinite Ito Integral 181
2. Continuous Martingales in LZ(P) 187
3. The Definite Ito Integral 189
4. Quadratic Variation 192
5. Ito's Formula and Two Applications 193
Problems 199
Notes 201
Appendix 203
1. Hilbert Spaces 203
2. Fourier Series 205
Bibliography 209
Index 217