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Description of Probability with Martingales

Probability theory is nowadays applied in a huge variety of fields including physics, engineering, biology, economics and the social sciences. This book is a modern, lively and rigorous account which has Doob's theory of martingales in discrete time as its main theme. It proves important results such as Kolmogorov's Strong Law of Large Numbers and the Three-Series Theorem by martingale techniques, and the Central Limit Theorem via the use of characteristic functions.

A distinguishing feature is its determination to keep the probability flowing at a nice tempo. It achieves this by being selective rather than encyclopaedic, presenting only what is essential to understand the fundamentals; and it assumes certain key results from measure theory in the main text. These measure-theoretic results are proved in full in appendices, so that the book is completely self-contained.

The book is written for students, not for researchers, and has evolved through several years of class testing. Exercises play a vital rle. 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.

Title Information

ISBN:

9780521406055

Pages:

272 pages

Format:

Paperback

Product Code:

14319

Publisher:

Cambridge University Press

Published:

14/02/1991

Edition:

1st Edition

Press and Industry Reviews

"… one of the best introductions to Martingale theory."
- Monatshefte für Mathematik

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Contents of Probability with Martingales

1. A branching-process example

Part A: Foundations

2. Measure spaces
3. Events
4. Random variables
5. Independence
6. Integration
7. Expectation
8. An easy strong law: product measure


Part B: 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 C: Characteristic functions

16. Basic properties of CFs
17. Weak convergence
18. The central limit theorem

Appendices
Exercises