Synopses & Reviews
The book is written for students, not for researchers, and has evolved through several years of class testing. Exercises play a vital role. Interesting and challenging problems, some with hints, consolidate what has already been learnt, and provide motivation to discover more of the subject than can be covered in a single introduction.
Review
"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." D.V. Feldman, Choice
Review
"...a nice textbook on measure-theoretic probability theory." Jia Gan Wang, Mathematical Reviews
Synopsis
This is a masterly introduction to the modern, and rigorous, theory of probability. The author emphasises martingales and develops all the necessary measure theory.
Synopsis
This work is an introduction to the modern, and rigorous, theory of probability. The author emphasises martingales and develops all the necessary measure theory.
Synopsis
'Probability theory is nowadays applied in a huge variety of fields from science and engineering to economics and politics. This is a modern, lively and rigorous introduction suitable for students that uses Doobâs theory of martingales in discrete time as its main theme. What distinguishes it from other books firstly is its selectivity - the author only includes what is essential to understand the fundamentals. He proves important results such as Kolmogorovâs Strong Law of Large Numbers and the Three-Series Theorem by martingale techniques, and the Central Limit Theorem via the use of characteristic functions. Secondly, the book is self-contained - the measure theory required is developed as needed, and then immediately exploited by applying it to real probability theory. More technical results are contained in appendices. Finally, the book is written for students, not researchers, and has evolved through several years of class testing. Exercises play a vital role. There is a full quota of interesting and challenging problems, some with hints, which aim to consolidate what the student has learned, and provide the challenge and motivation to learn much more about the subject than can be covered in a single introduction.\n
'
Synopsis
The author adopts the martingale theory as his main theme in this introduction to the modern theory of probability, which is, perhaps, at a practical level, one of the most useful mathematical theories ever devised.
Description
Includes bibliographical references (p. 243-245) and index.
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.