Probability with Martingales

Produktbeschreibung

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 rôle. 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. 

Inhaltsverzeichnis

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. 

Kritik

'... one of the best introductions to Martingale theory.' Monatshefte für Mathematik  

Autoreninfo

Disclaimer. We have made every effort to make certain that all information is factually correct, comprehensive, and up-to-date. However, this book should not be used as a substitute for the knowledge and expertise of a licensed healthcare professional. You should always consult your doctor or other healthcare professionals before taking any medication. The drug information contained herein is subject to change and is not intended to cover all possible uses, directions, precautions, warnings, drug interactions, allergic reactions, or adverse effects. The absence of warnings or other information for a given drug does not indicate that the drug or drug combination is safe, effective, or appropriate for all patients or all specific uses.