Preface vii
Chapter 1. Measure Theory 1
1.1. Introduction 1
1.2. Construction of Measures 3
1.3. Integration 7
1.4. Transformations 13
1.5. Product Spaces 14
1.6. Distributions and Expectations 16
Chapter 2. Weak Convergence 19
2.1. Characteristic Functions 19
2.2. Moment-Generating Functions 22
2.3. Weak Convergence 24
Chapter 3. Independent Sums 35
3.1. Independence and Convolution 35
3.2. Weak Law of Large Numbers 37
3.3. Strong Limit Theorems 40
3.4. Series of Independent Random Variables 43
3.5. Strong Law of Large Numbers 48
3.6. Central Limit Theorem 49
3.7. Accompanying Laws 54
3.8. Infinitely Divisible Distributions 59
3.9. Laws of the Iterated Logarithm 66
Chapter 4. Dependent Random Variables 73
4.1. Conditioning 73
4.2. Conditional Expectation 79
4.3. Conditional Probability 81
4.4. Markov Chains 84
4.5. Stopping Times and Renewal Times 89
4.6. Countable State Space 90
4.7. Some Examples 98
Chapter 5. Martingales 109
5.1. Definitions and Properties 109
5.2. Martingale Convergence Theorems 112
v
CONTENTS
5.3. Doob Decomposition Theorem 115
5.4. Stopping Times 117
5.5. Up-crossing Inequality 120
5.6. Martingale Transforms, Option Pricing 121
5.7. Martingales and Markov Chains 123
Chapter 6. Stationary Stochastic Processes 131
6.1. Ergodic Theorems 131
6.2. Structure of Stationary Measures 135
6.3. Stationary Markov Processes 137
6.4. Mixing Properties of Markov Processes 141
6.5. Central Limit Theorem for Martingales 143
6.6. Stationary Gaussian Processes 147
Chapter 7. Dynamic Programming and Filtering 157
7.1. Optimal Control 157
7.2. Optimal Stopping 158
7.3. Filtering 161
Bibliography 163
Index 165