A Course in Probability Theory, Third Edition

Cover......Page 1
Course in Probability Theory......Page 3
Copyright......Page 4
Contents......Page 5
Preface to the third edition......Page 8
Preface to the second edition......Page 10
Preface to the first edition......Page 12
1 Distribution function......Page 16
2.1 Classes of sets......Page 30
2.2 Probability measures and their distribution functions......Page 35
3.1 General definitions......Page 48
3.2 Properties of mathematical expectation......Page 55
3.3 Independence......Page 67
4.1 Various modes of convergence......Page 82
4.2 Almost sure convergence; Borel-Cantelli lemma......Page 89
4.3 Vague convergence......Page 98
4.4 Continuation......Page 105
4.5 Uniform integrability; convergence of moments......Page 113
5.1 Simple limit theorems......Page 120
5.2 Weak law of large numbers......Page 126
5.3 Convergence of series......Page 135
5.4 Strong law of large numbers......Page 143
5.5 Applications......Page 152
6.1 General properties; convolutions......Page 164
6.2 Uniqueness and inversion......Page 174
6.3 Convergence theorems......Page 183
6.4 Simple applications......Page 189
6.5 Representation theorems......Page 201
6.6 Multidimensional case ; Laplace transforms......Page 210
7.1 Liapounov\'s theorem......Page 219
7.2 Lindeberg-Feller theorem......Page 228
7.3 Ramifications of the central limit theorem......Page 238
7.4 Error estimation......Page 249
7.5 Law of the iterated logarithm......Page 256
7.6 Infinite divisibility......Page 264
8.1 Zero-or-one laws......Page 277
8.2 Basic notions......Page 284
8.3 Recurrence......Page 292
8.4 Fine structure......Page 302
8.5 Continuation......Page 312
9.1 Basic properties of conditional expectation......Page 324
9.2 Conditional independence ; Markov property......Page 336
9.3 Basic properties of smartingales......Page 348
9.4 Inequalities and convergence......Page 360
9.5 Applications......Page 374
1 Construction of measure......Page 389
2 Characterization of extensions......Page 394
3 Measures in R......Page 401
4 Integral......Page 409
5 Applications......Page 421
3. Probability theory......Page 427
5. Supplementary reading......Page 428
Index......Page 429