## The Annals of Probability

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

### Optimum Bounds for the Distributions of Martingales in Banach Spaces

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176988477

**Digital Object Identifier**

doi:10.1214/aop/1176988477

**Mathematical Reviews number (MathSciNet)**

MR1331198

**Zentralblatt MATH identifier**

0836.60015

**JSTOR**

links.jstor.org

**Subjects**

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

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aop/1176988477