We develop a new method to study the tails of a sum of independent mean zero Banach-space valued random variables $(X_i)_{i \leq N}.$ It relies on a new isoperimetric inequality for subsets of a product of probability spaces. In particular, we prove that for $p \geq 1,$ $\bigg\|\sum_{i \leq N} X_i\bigg\|_p \leq \frac{Kp}{1 + \log p}\bigg(\bigg\|\sum_{i \leq N} X_i\bigg\|_1 + \|\max_{i \leq N}\|X_i\|\|_p\bigg),$ where $K$ is a universal constant. Other optimal inequalities for exponential moments are obtained.
@article{1176991174,
author = {Talagrand, Michel},
title = {Isoperimetry and Integrability of the Sum of Independent Banach-Space Valued Random Variables},
journal = {Ann. Probab.},
volume = {17},
number = {4},
year = {1989},
pages = { 1546-1570},
language = {en},
url = {http://dml.mathdoc.fr/item/1176991174}
}
Talagrand, Michel. Isoperimetry and Integrability of the Sum of Independent Banach-Space Valued Random Variables. Ann. Probab., Tome 17 (1989) no. 4, pp. 1546-1570. http://gdmltest.u-ga.fr/item/1176991174/