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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
|---|---|---|---|---|---|
Searching... | 30000010337192 | QA274.5 W55 1991 | Open Access Book | Book | Searching... |
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Summary
Summary
This is a masterly introduction to the modern and rigorous theory of probability. The author adopts the martingale theory as his main theme and moves at a lively pace through the subject's rigorous foundations. Measure theory is introduced and then immediately exploited by being applied to real probability theory. Classical results, such as Kolmogorov's Strong Law of Large Numbers and Three-Series Theorem are proved by martingale techniques. A proof of the Central Limit Theorem is also given. The author's style is entertaining and inimitable with pedagogy to the fore. Exercises play a vital role; there is a full quota of interesting and challenging problems, some with hints.
Reviews 1
Choice Review
If one is teaching advanced mathematics to capable students, one plans to spend time satisfying requests for examples and counterexamples, for additional intuition and clarification, for alternative routes, for historical contextual applications, and for simple reminders of glimpses of the ^D["big picture.^D]" Such material is often sacrificed in the transition to formal exposition, presumably in the service of that false god, logical flow. Williams, who writes as though he were reading the reader's mind, does a brilliant job of leaving it all in. And well that he does, since the bridge from basic probability theory to measure theoretic probability can be difficult crossing. Indeed, so lively is the development from scratch of the needed measure theory, that students of real analysis, even those with no special interest in probability, should take note. The prominence throughout given to the problems, subtleties, and pathologies that compel new abstractions helps the reader maintain connections to basic intuition even as it opens new vistas. Recommended. Undergraduate; graduate.
Table of Contents
| 1 A branching-process example |
| Part I Foundations |
| 2 Measure spaces |
| 3 Events |
| 4 Random variables |
| 5 Independence |
| 6 Integration |
| 7 Expectation |
| 8 An easy strong law: product measure |
| Part II Martingale Theory |
| 9 Conditional expectation |
| 10 Martingales |
| 11 The convergence theorem |
| 12 Martingales bounded in L2 |
| 13 Uniform integrability |
| 14 UI martingales |
| 15 Applications |
| Part III Characteristic Functions |
| 16 Basic properties of CFs |
| 17 Weak convergence |
| 18 The central limit theorem |
| Appendices |
| Exercises |
