1 | Distribution function
1.1 Monotone functions 1
1.2 Distribution functions 7
1.3 Absolutely continuous and singular distributions 11
2 | Measure theory
2.1 Classes of sets 16
2.2 Probability measures and their distribution
functions 21
3 | Random variable. Expectation. Independence
3.1 General definitions 34
3.2 Properties of mathematical expectation 41
3.3 Independence 53
4 I Convergence concepts
4.1 Various modes of convergence 68
4.2 Almost sure convergence; Borel-Cantelli lemma 75
4.3 Vague convergence 84
4.4 Continuation 91
4.5 Uniform integrability; convergence of moments 99
5 Law of large numbers. Random series
5.1 Simple limit theorems 106
5.2 Weak law of large numbers 112
5.3 Convergence of series 121
5.4 Strong law of large numbers 129
5.5 Applications 138
Bibliographical Note 148
Characteristic function
6.1 General properties; convolutions 150
6.2 Uniqueness and inversion 160
6.3 Convergence theorems 169
6.4 Simple applications 175
6.5 Representation theorems 187
6.6 Multidimensional case; Laplace transforms 196
Bibliographical Note 204
Central limit theorem and its ramifications
7.1 Liapounov's theorem 205
7.2 Lindeberg-Feller theorem 214
7.3 Ramifications of the central limit theorem 224
7.4 Error estimation 235
7.5 Law of the iterated logarithm 242
7.6 Infinite divisibility 250
Bibliographical Note 261
8 | Random walk
8.1 Zero-or-one laws 263
8.2 Basic notions 270
8.3 Recurrence 278
8.4 Fine structure 288
8.5 Continuation 298
Bibliographical Note 308
9 Conditioning. Markov property. Martingale
9.1 Basic properties of conditional expectation 310
9.2 Conditional independence; Markov property 322
9.3 Basic properties of smartingales 334
9.4 Inequalities and convergence 346
9.5 Applications 360
Bibliographical Note 373
| Supplement: Measure and Integral
1 Construction of measure 375
2 Characterization of extensions 380
3 Measures in R 387
4 Integral 395
5 Applications 407
General Bibliography 413
Index 415