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Preface vii
I Random Maps, Distribution, and Mathematical Expectation 1
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
II Independence, Conditional Expectation 19
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
III Martingales and Stopping Times 37
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
IV Classical Zero–One Laws, Laws of Large Numbers and Deviations 49
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
V Weak Convergence of Probability Measures 59
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
VI Fourier Series, Fourier Transform, and Characteristic Functions 73
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
VII Classical Central Limit Theorems 99
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
VIII Laplace Transforms and Tauberian Theorem 107
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
xii CONTENTS
IX Random Series of Independent Summands 121
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
X Kolmogorov’s Extension Theorem and Brownian Motion 129
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
XI Brownian Motion: The LIL and Some Fine-Scale Properties 141
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
XII Skorokhod Embedding and Donsker’s Invariance Principle 147
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
XIII A Historical Note on Brownian Motion 167
A Measure and Integration 171
1 Measures and the Carath´eodory Extension. . . . . . . . . . . . . 171
2 Integration and Basic Convergence Theorems . . . . . . . . . . 176
3 Product Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4 Riesz Representation on C(S) . . . . . . . . . . . . . . . . . . . 183
B Topology and Function Spaces 187
C Hilbert Spaces and Applications in Measure Theory 193
1 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
2 Lebesgue Decomposition and the Radon–Nikodym Theorem . . 196
References 201
Index 205
Symbol Index 211
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