A Course in Probability Theory, Revised Edition
by Kai Lai ChungFormat: Nonspecific Binding
Pub. Date: 2000-10-17
Publisher(s): Elsevier S & T
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Summary
This book contains about 500 exercises consisting mostly of special cases and examples, second thoughts and alternative arguments, natural extensions, and some novel departures. With a few obvious exceptions they are neither profound nor trivial, and hints and comments are appended to many of them. If they tend to be somewhat inbred, at least they are relevant to the text and should help in its digestion. As a bold venture I have marked a few of them with a * to indicate a "must", although no rigid standard of selection has been used. Some of these are needed in the book, but in any case the reader's study of the text will be more complete after he has tried at least those problems.
Table of Contents
| Preface to the third edition | p. ix |
| Preface to the second edition | p. xi |
| Preface to the first edition | p. xiii |
| Distribution function | |
| Monotone functions | p. 1 |
| Distribution functions | p. 7 |
| Absolutely continuous and singular distributions | p. 11 |
| Measure theory | |
| Classes of sets | p. 16 |
| Probability measures and their distribution functions | p. 21 |
| Random variable. Expectation. Independence | |
| General definitions | p. 34 |
| Properties of mathematical expectation | p. 41 |
| Independence | p. 53 |
| Convergence concepts | |
| Various modes of convergence | p. 68 |
| Almost sure convergence; Borel-Cantelli lemma | p. 75 |
| Vague convergence | p. 84 |
| Continuation | p. 91 |
| Uniform integrability; convergence of moments | p. 99 |
| Law of large numbers. Random series | |
| Simple limit theorems | p. 106 |
| Weak law of large numbers | p. 112 |
| Convergence of series | p. 121 |
| Strong law of large numbers | p. 129 |
| Applications | p. 138 |
| Bibliographical Note | p. 148 |
| Characteristic function | |
| General properties; convolutions | p. 150 |
| Uniqueness and inversion | p. 160 |
| Convergence theorems | p. 169 |
| Simple applications | p. 175 |
| Representation theorems | p. 187 |
| Multidimensional case; Laplace transforms | p. 196 |
| Bibliographical Note | p. 204 |
| Central limit theorem and its ramifications | |
| Liapounov's theorem | p. 205 |
| Lindeberg-Feller theorem | p. 214 |
| Ramifications of the central limit theorem | p. 224 |
| Error estimation | p. 235 |
| Law of the iterated logarithm | p. 242 |
| Infinite divisibility | p. 250 |
| Bibliographical Note | p. 261 |
| Random walk | |
| Zero-or-one laws | p. 263 |
| Basic notions | p. 270 |
| Recurrence | p. 278 |
| Fine structure | p. 288 |
| Continuation | p. 298 |
| Bibliographical Note | p. 308 |
| Conditioning. Markov property. Martingale | |
| Basic properties of conditional expectation | p. 310 |
| Conditional independence; Markov property | p. 322 |
| Basic properties of smartingales | p. 334 |
| Inequalities and convergence | p. 346 |
| Applications | p. 360 |
| Bibliographical Note | p. 373 |
| Supplement: Measure and Integral | |
| Construction of measure | p. 375 |
| Characterization of extensions | p. 380 |
| Measures in R | p. 387 |
| Integral | p. 395 |
| Applications | p. 407 |
| General Bibliography | p. 413 |
| Index | p. 415 |
| Table of Contents provided by Syndetics. All Rights Reserved. |
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