Annals of Probability

Optimum Bounds for the Distributions of Martingales in Banach Spaces

Iosif Pinelis

Full-text: Open access


A general device is proposed, which provides for extension of exponential inequalities for sums of independent real-valued random variables to those for martingales in the 2-smooth Banach spaces. This is used to obtain optimum bounds of the Rosenthal-Burkholder and Chung types on moments of the martingales in 2-smooth Banach spaces. In turn, it leads to best-order bounds on moments of sums of independent random vectors in any separable Banach spaces. Although the emphasis is put on infinite-dimensional martingales, most of the results seem to be new even for one-dimensional martingales. Moreover, the bounds on moments of the Rosenthal-Burkholder type seem to be to a certain extent new even for sums of independent real-valued random variables. Analogous inequalities for (one-dimensional) supermartingales are given.

Article information

Ann. Probab., Volume 22, Number 4 (1994), 1679-1706.

First available in Project Euclid: 19 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 60G42: Martingales with discrete parameter 60G50: Sums of independent random variables; random walks 60F10: Large deviations

Distribution inequalities exponential inequalities bounds on moments martingales in Banach spaces 2-smooth Banach spaces sums of independent random variables


Pinelis, Iosif. Optimum Bounds for the Distributions of Martingales in Banach Spaces. Ann. Probab. 22 (1994), no. 4, 1679--1706. doi:10.1214/aop/1176988477.

Export citation