A general bounded law of the iterated logarithm for Banach space valued random variables is established. Our results implies: (a) the bounded LIL of Ledoux and Talagrand, (b) a bounded LIL for random variables in the domain of attraction of a Gaussian law and (c) new LIL results for random variables outside the domain of attraction of a Gaussian law in cases where the classical norming sequence $\{\sqrt{nLLn}\}$ does not work. Basic ingredients of our proof are an infinite-dimensional Fuk-Nagaev type inequality and an infinite-dimensional version of Klass's $K$-function.
Publié le : 1993-10-14
Classification:
Bounded law of the iterated logarithm,
$K$-function,
LIL behavior,
randomization,
Rademacher random variables,
60F15,
60B12
@article{1176989009,
author = {Einmahl, Uwe},
title = {Toward a General Law of the Iterated Logarithm in Banach Space},
journal = {Ann. Probab.},
volume = {21},
number = {4},
year = {1993},
pages = { 2012-2045},
language = {en},
url = {http://dml.mathdoc.fr/item/1176989009}
}
Einmahl, Uwe. Toward a General Law of the Iterated Logarithm in Banach Space. Ann. Probab., Tome 21 (1993) no. 4, pp. 2012-2045. http://gdmltest.u-ga.fr/item/1176989009/