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 $p/\ln p$.
Publié le : 1990-10-14
Classification:
Martingale,
moment inequalities,
good $\lambda$ inequality,
60E15,
60G42
@article{1176990639,
author = {Hitczenko, Pawel},
title = {Best Constants in Martingale Version of Rosenthal's Inequality},
journal = {Ann. Probab.},
volume = {18},
number = {4},
year = {1990},
pages = { 1656-1668},
language = {en},
url = {http://dml.mathdoc.fr/item/1176990639}
}
Hitczenko, Pawel. Best Constants in Martingale Version of Rosenthal's Inequality. Ann. Probab., Tome 18 (1990) no. 4, pp. 1656-1668. http://gdmltest.u-ga.fr/item/1176990639/