## Electronic Journal of Probability

### Sharp inequalities for martingales with values in $\ell_\infty^N$

Adam Osękowski

#### Abstract

The objective of the paper is to study sharp inequalities for transforms of martingales taking values in $\ell_\infty^N$. Using Burkholder's method combined with an intrinsic duality argument, we identify, for each $N\geq 2$, the best constant $C_N$ such that the following holds. If $f$ is a martingale with values in $\ell_\infty^N$ and $g$ is its transform by a sequence of signs, then

$$||g||_1\leq C_N ||f||_\infty.$$

This is closely related to the characterization of UMD spaces in terms of the so-called $\eta$ convexity, studied in the eighties by Burkholder and Lee.

#### Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 73, 19 pp.

Dates
Accepted: 9 August 2013
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465064298

Digital Object Identifier
doi:10.1214/EJP.v18-2667

Mathematical Reviews number (MathSciNet)
MR3091719

Zentralblatt MATH identifier
1287.60052

Subjects
Primary: 60G42: Martingales with discrete parameter
Secondary: 60G44: Martingales with continuous parameter

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

#### Citation

Osękowski, Adam. Sharp inequalities for martingales with values in $\ell_\infty^N$. Electron. J. Probab. 18 (2013), paper no. 73, 19 pp. doi:10.1214/EJP.v18-2667. https://projecteuclid.org/euclid.ejp/1465064298

#### References

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