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Preface to the third edition iii
Preface to the second edition v
Preface to the first edition vii
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 function 21
3 Random variable, Expectation.Independence
3.1 General definition 34
3.2 Properties of mathematical expectation 41
3.3 Independence 53
4 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 untegrability; convergence of moments 99
5 Law of large numbers, Randrom series
5.1 Simple limit theorems 106
5.2 Weak low of large nymbers 112
5.3 Convergence of serices 121
5.4 Strong law of large numbers 129
5.5 Applications 138
Bibliographical Note 148
6 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 Multidimentstional case; Laplace transforms 196
Bibliographical Note 204
7 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 divistibility 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 propery 322
9.3 Basci properties of smartingales 334
9.4 Inequalities and convergence 346
9.5 Applications 360
Bibliographical Note 373
Supplement: Measure and Integral
1 Construvtion of measure 375
2 Characterization of extensions 380
3 Measures in R 387
4 Integral 395
5 Applications 407
General Bibliography 413
Index 415