## The Annals of Probability

### Best Constants in Martingale Version of Rosenthal's Inequality

Pawel Hitczenko

#### Abstract

The following generalization of Rosenthal's inequality was proved by Burkholder: $A^{-1}_p\{\|s(f)\|_p + \|d^\ast\|_p\} \leq \|f^\ast\|_p \leq B_p\{\|s(f)\|_p + \|d^\ast\|_p\},$ for all martingales $(f_n)$. It is known that $A_p$ grows like $\sqrt{p}$ as $p \rightarrow \infty$. In this paper we prove that the growth rate of $B_p$ as $p \rightarrow \infty$ is $p/\ln p$.

#### Article information

Source
Ann. Probab. Volume 18, Number 4 (1990), 1656-1668.

Dates
First available in Project Euclid: 19 April 2007

http://projecteuclid.org/euclid.aop/1176990639

JSTOR

Digital Object Identifier
doi:10.1214/aop/1176990639

Mathematical Reviews number (MathSciNet)
MR1071816

Zentralblatt MATH identifier
0725.60018

Subjects
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60G42: Martingales with discrete parameter

#### Citation

Hitczenko, Pawel. Best Constants in Martingale Version of Rosenthal's Inequality. The Annals of Probability 18 (1990), no. 4, 1656--1668. doi:10.1214/aop/1176990639. http://projecteuclid.org/euclid.aop/1176990639.