Martingale Inequalities in Discrete Time

Abstract

This part of the monograph contains a study of inequalities for discrete-time martingales, both in the Hilbert-space and Banach-space-valued setting. It is worth stressing here that the setting of Hilbert-space-valued differentially subordinate martingales has been studied intensively and, essentially, the sharp versions of all the crucial estimates arising in this context have been successfully proved. On the other hand, much is to be done in the non-Hilbert case, in which no optimal constants are known.