A little book of martingales (Q6595957)

This book covers the main results on discrete-time martingales and gives some applications of these results. It consists of eight chapters. The first three chapters provide the necessary background: monotone class theorem, measure and its properties, measurable functions and integration, probability distributions, characteristic functions, signed measures, conditional expectations, uniform integrability and some others. Martingales appear already in Chapter 4. The authors define stopping times, which are fundamental in martingale theory, and prove Doob's optional sampling theorem. Then Wald identities and Doob's maximal inequality are presented. \N\NChapter 5 is devoted to convergence in \(L^p\) and almost sure convergence results for martingales, sub- and super-martingales. Upcrossing lemma is a starting point here. The notion of time-reversed martingale is also introduced. Chapter 6 gives some applications of the results from Chapter 5. It is shown how the Kolmogorov 0-1 law and the Hewitt-Savage 0-1 law follow from the reverse martingale convergence theorem. The strong laws of large numbers for averages of independent and identically distributed random variables, as well as for U-statistics, are proved by using reverse martingales. The strong law for exchangeable sequences is also established, as well as de-Finetti's theorem for exchangeable random variables. The final topic in this chapter is Kakutani's theorem for product martingales. \N\NChapter 7 is devoted to the central limit theorem (CLT) for martingales. As illustrations, it is applied to a simple urn model, to the trace of a random matrix, and to Markov chains. In Chapter 8 some additional topics such as forward martingale representation for U-statistics, extended/conditional Borel-Cantelli lemma, Azuma-Hoeffding inequality, conditional three series theorem, strong and weak laws of large numbers for martingales, and the Kesten-Stigum theorem for a simple branching process are covered. The so-called Burkholder inequalities are also considered. \N\NThe book corresponds to a master's course in this subject. It is very clearly and interestingly written, has quite modern content, the proofs are given rigorously, but with maximum simplicity. The book will be useful and can be recommended for students, postgraduates, teachers and anyone who has decided to learn or refresh their memory on the theory of random processes, as well as for practitioners interested in applications of martingale theory.