This concise textbook, fashioned along the syllabus for master's and Ph.D. programmes, covers basic results on discrete-time martingales and applications. It includes additional interesting and useful topics, providing the ability to move beyond. 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 HewittSavage, the strong laws for U-statistics and 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. Additional topics include forward martingale representation for U-statistics, conditional BorelCantelli lemma, AzumaHoeffding inequality, conditional three series theorem, strong law for martingales and the KestenStigum theorem for a simple branching process. The prerequisite for this course is a first course in measure theoretic probability. The book recollects its essential concepts and results, mostly without proof, but full details have been provided for the RadonNikodym theorem and the concept of conditional expectation.

Offers a thorough exploration of discrete-time martingale theory, suitable for master's students Presents complex concepts in a clear and understandable manner with examples and exercises for active engagement Emphasizes applications in different areas such as urn models, CLT, and branching processes

Autorentext

Arup Bose is an Honorary Visiting Professor at the Indian Statistical Institute since his superannuation in 2024. He has published more than 150 research articles in probability, statistics, econometrics and economics., as well as six books (singly or with others) covering topics in random matrices, non-commutative probability, U-statistics, Mm estimates, resampling, and martingales. He is a Fellow of the Institute of Mathematical Statistics, the Indian National Science Academy, the National Academy of Science and the Indian Academy of Sciences. He has won the Shanti Swarup Bhatnagar Prize and the C.R. Rao award from the Governemtn of India, and the Mahalanobis International Award for Lifetime Achievements from the International Statistical Institute.

Inhalt

Measure.- Signed measure.- Conditional expectation.- Martingales.- Almost sure and Lp convergence.- Application of convergence theorems.- Central limit theorem.- Additional Topics.