**Aims & Learning Objectives:**
**Aims: **To stimulate through theory and especially examples, an interest and appreciation of the power of this elegant method in analysis and probability. Applications of the theory are at the heart of this course.
**Objectives: **
By the end of the course, students should be familiar with the main results and techniques of discrete time martingale theory. They will have seen applications of martingales in proving some important results from classical probability theory, and they should be able to recognise and apply martingales in solving a variety of more elementary problems.
**Content: ** Topics will be chosen from the following:
Review of fundamental concepts. Conditional expectation. Martingales, stopping times, Optional-Stopping Theorem. The Convergence Theorem. L² -bounded martingales, the random-signs problem. Angle-brackets process, Lévy's Borel-Cantelli Lemma. Uniform integrability. UI martingales, the "Downward" Theorem, the Strong Law, the Submartingale Inequality. Likelihood ratio, Kakutani's theorem.
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