A Course in Probability Theory, Revised Edition
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Table of contents
- Title Pageiii
- Copyright Pageiv
- Contentsv
- Preface to the third editionix
- Preface to the second editionxi
- Preface to the first editionxiii
- Chapter 1. Distribution function1
- 1.1 Monotone functions1
- 1.2 Distribution functions7
- 1.3 Absolutely continuous and singular distributions11
- Chapter 2. Measure theory16
- 2.1 Classes of sets16
- 2.2 Probability measures and their distribution functions21
- Chapter 3. Random variable. Expectation. Independence34
- 3.1 General definitions34
- 3.2 Properties of mathematical expectation41
- 3.3 Independence53
- Chapter 4. Convergence concepts68
- 4.1 Various modes of convergence68
- 4.2 Almost sure convergence; Borel–Cantelli lemma75
- 4.3 Vague convergence84
- 4.4 Continuation91
- 4.5 Uniform integrability; convergence of moments99
- Chapter 5. Law of large numbers. Random series106
- 5.1 Simple limit theorems106
- 5.2 Weak law of large numbers112
- 5.3 Convergence of series121
- 5.4 Strong law of large numbers129
- 5.5 Applications138
- Bibliographical Note148
- Chapter 6. Characteristic function150
- 6.1 General properties; convolutions150
- 6.2 Uniqueness and inversion160
- 6.3 Convergence theorems169
- 6.4 Simple applications175
- 6.5 Representation theorems187
- 6.6 Multidimensional case; Laplace transforms196
- Bibliographical Note204
- Chapter 7. Central limit theorem and its ramifications205
- 7.1 Liapounov’s theorem205
- 7.2 Lindeberg–Feller theorem214
- 7.3 Ramifications of the central limit theorem224
- 7.4 Error estimation235
- 7.5 Law of the iterated logarithm242
- 7.6 Infinite divisibility250
- Bibliographical Note261
- Chapter 8. Random walk263
- 8.1 Zero-or-one laws263
- 8.2 Basic notions270
- 8.3 Recurrence278
- 8.4 Fine structure288
- 8.5 Continuation298
- Bibliographical Note308
- Chapter 9. Conditioning. Markov property. Martingale310
- 9.1 Basic properties of conditional expectation310
- 9.2 Conditional independence; Markov property322
- 9.3 Basic properties of smartingales334
- 9.4 Inequalities and convergence346
- 9.5 Applications360
- Bibliographical Note373
- Supplement: Measure and Integral375
- 1 Construction of measure375
- 2 Characterization of extensions380
- 3 Measures in R387
- 4 Integral395
- 5 Applications407
- General Bibliography413
- Index415
Since the publication of the first edition of this classic textbook over thirty years ago, tens of thousands of students have used A Course in Probability Theory. New in this edition is an introduction to measure theory that expands the market, as this treatment is more consistent with current courses.
While there are several books on probability, Chung's book is considered a classic, original work in probability theory due to its elite level of sophistication.
While there are several books on probability, Chung's book is considered a classic, original work in probability theory due to its elite level of sophistication.