Let $\{\xi_k, k \geqslant 1\}$ be a sequence of random variables uniformly distributed over $\lbrack 0, 1\rbrack$ and let $\mathscr{F}$ be a class of functions on $\lbrack 0, 1\rbrack$ with $\int^1_0 f(x) dx = 0$. In this paper we give upper and lower bounds for $\sup_{f \in \mathscr{F}}|\sigma_{k \leqslant N}f(\xi_k)|$ for the class of functions of variation bounded by 1 and for the class of functions satisfying a Lipschitz condition.
Publié le : 1978-12-14
Classification:
Law of the iterated logarithm,
mixing sequences of random variables,
lacunary sequences,
Hilbert space valued random variables,
60F15,
42A44,
10K15
@article{1176995385,
author = {Kaufman, R. and Philipp, Walter},
title = {A Uniform Law of the Iterated Logarithm for Classes of Functions},
journal = {Ann. Probab.},
volume = {6},
number = {6},
year = {1978},
pages = { 930-952},
language = {en},
url = {http://dml.mathdoc.fr/item/1176995385}
}
Kaufman, R.; Philipp, Walter. A Uniform Law of the Iterated Logarithm for Classes of Functions. Ann. Probab., Tome 6 (1978) no. 6, pp. 930-952. http://gdmltest.u-ga.fr/item/1176995385/