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 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.
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Excelent book for a probability course at postgraduate level.
Have you ever wondered why in statistics you were given theorems of probability to use without ever really being told where these came from? After reading this book you’ll understand why they didn’t tell you: about 70 pages of equations into the book they finally get to defining p(X|Y). So be warned, it’s a text book written for mathematicians, which makes sense given that generally only mathematicians need to understand probability theory in this much technical depth. Its main limitation is that it doesn’t cover stochastic processes in continuous time. However, once you understand the theory of martigales well in discrete time, going to continuous time isn’t too much of a step