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.
Publié le : 1994-10-14
Classification:
Distribution inequalities,
exponential inequalities,
bounds on moments,
martingales in Banach spaces,
2-smooth Banach spaces,
sums of independent random variables,
60E15,
60B12,
60G42,
60G50,
60F10
@article{1176988477,
author = {Pinelis, Iosif},
title = {Optimum Bounds for the Distributions of Martingales in Banach Spaces},
journal = {Ann. Probab.},
volume = {22},
number = {4},
year = {1994},
pages = { 1679-1706},
language = {en},
url = {http://dml.mathdoc.fr/item/1176988477}
}
Pinelis, Iosif. Optimum Bounds for the Distributions of Martingales in Banach Spaces. Ann. Probab., Tome 22 (1994) no. 4, pp. 1679-1706. http://gdmltest.u-ga.fr/item/1176988477/