{"product_id":"probability-with-martingales-paperback-9780521406055","title":"Probability with Martingales (Paperback) 9780521406055","description":"\u003cfont face=\"Georgia\"\u003e\r\n\u003cp\u003e\u003cfont size=\"6\"\u003eProbability with Martingales\u003c\/font\u003e\u003cbr\u003e\r\n\r\n\r\n\u003c\/p\u003e\n\u003cp\u003e\u003cem\u003eThis is a masterly introduction to the modern, and rigorous, theory of probability. The author emphasises martingales and develops all the necessary measure theory.\u003c\/em\u003e\u003c\/p\u003e\r\n\r\n\r\n\u003cp\u003e\u003cfont size=\"4\"\u003eDavid Williams (Author)\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e9780521406055, Cambridge University Press\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003ePaperback, published 14 February 1991\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e265 pages, 3 b\/w illus.\u003cbr\u003e22.8 x 15.2 x 1.7 cm, 0.412 kg\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\r\n\u003cp align=\"justify\"\u003e\u003cem\u003e\u003cfont size=\"3\"\u003e'… one of the best introductions to Martingale theory.'   Monatshefte für Mathematik\u003c\/font\u003e\u003c\/em\u003e\u003c\/p\u003e\r\n\r\n\u003cp align=\"justify\"\u003e\u003cstrong\u003e\u003cfont size=\"3\"\u003eProbability 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.\u003c\/font\u003e\u003c\/strong\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e1. A branching-process example\u003cbr\u003e Part I. Foundations: 2. Measure spaces\u003cbr\u003e 3. Events\u003cbr\u003e 4. Random variables\u003cbr\u003e 5. Independence\u003cbr\u003e 6. Integration\u003cbr\u003e 7. Expectation\u003cbr\u003e 8. An easy strong law: product measure\u003cbr\u003e Part II. Martingale Theory: 9. Conditional expectation\u003cbr\u003e 10. Martingales\u003cbr\u003e 11. The convergence theorem\u003cbr\u003e 12. Martingales bounded in L2\u003cbr\u003e 13. Uniform integrability\u003cbr\u003e 14. UI martingales\u003cbr\u003e 15. Applications\u003cbr\u003e Part III. Characteristic Functions: 16. Basic properties of CFs\u003cbr\u003e 17. Weak convergence\u003cbr\u003e 18. The central limit theorem\u003cbr\u003e Appendices\u003cbr\u003e Exercises.\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003eSubject Areas: Probability \u0026amp; statistics [\u003ca title=\"See our other books on Probability \u0026amp; statistics\" href=\"https:\/\/freshlyprintedbooks.co.uk\/search?q=%22Probability%20\u0026amp;%20statistics%20%5BPBT%5D%22\"\u003ePBT\u003c\/a\u003e]\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\u003c\/font\u003e","brand":"Cambridge University Press","offers":[{"title":"Default Title","offer_id":46000372842776,"sku":"9780521406055","price":35.99,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0730\/2037\/5320\/products\/9780521406055i_11d41bf3-75cd-414b-842a-b1c8ddfa1f7a.jpg?v=1691381216","url":"https:\/\/freshlyprintedbooks.co.uk\/products\/probability-with-martingales-paperback-9780521406055","provider":"Freshly Printed Books","version":"1.0","type":"link"}