A Little Book of Martingales

Contents: 1. Measure. 2. Signed measure. 3. Conditional expectation. 4. Martingales. 5. Almost sure and Lp convergence. 6. Application of convergence theorems. 7. Central limit theorem. 8. Additional Topics.

This little book, suitable for masters and Ph.D. programs, covers basic results on discrete time martingales and their applications. It includes some additional interesting and useful topics. Adequate details are provided, with exercises within the text and at the end of chapters.

Basic results include Doob's optional sampling theorem, Wald identities, Doob's maximal inequality, upcrossing lemma, time-reversed martingales, a variety of convergence results, and a limited discussion of the Burkholder inequalities. Applications include the 0-1 laws of Kolmogorov and Hewitt-Savage, the strong laws for U-statistics and for exchangeable sequences, de-Finetti's theorem for exchangeable sequences, and Kakutani's theorem for product martingales. A simple central limit theorem for martingales is proven and applied to a basic urn model, the trace of a random matrix, and Markov chains. Some of the additional topics covered are forward martingale representation for U-statistics, conditional Borel-Cantelli lemma, Azuma-Hoeffding inequality, conditional three series theorem, strong law for martingales, and the Kesten-Stigum theorem for a simple branching process.

A first course in measure theoretic probability is a prerequisite. We have recollected its essential concepts and results, mostly without proofs.