Let $X_1, X_2, \cdots$ be independent random variables with values in a Banach space $E$. It is then shown that Chung's version of the strong law of large numbers holds, if and only if $E$ is of type $p$. If the $X_n$'s are identically distributed, then it is shown that the central limit theorem is valid, if and only if $E$ is of type 2. Similar results are obtained for vectorvalued martingales.
Publié le : 1976-08-14
Classification:
Central limit theorem,
law of large numbers,
Banach space valued random variables,
martingales,
Banach space type,
modulus of uniform smoothness,
60F05,
60B10,
46E15
@article{1176996029,
author = {Hoffmann-Jorgensen, J. and Pisier, G.},
title = {The Law of Large Numbers and the Central Limit Theorem in Banach Spaces},
journal = {Ann. Probab.},
volume = {4},
number = {6},
year = {1976},
pages = { 587-599},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996029}
}
Hoffmann-Jorgensen, J.; Pisier, G. The Law of Large Numbers and the Central Limit Theorem in Banach Spaces. Ann. Probab., Tome 4 (1976) no. 6, pp. 587-599. http://gdmltest.u-ga.fr/item/1176996029/