Let $(X_n)_{n\geq 1}$ be a sequence of i.i.d. r.v.'s with values in a measurable space $(E, \mathscr{E})$ of law $\mu$, and consider the empirical process $L_n(f) = (1/n)\sum^n_{k=1} f(X_k)$ with $f$ varying in a class of bounded functions $\mathscr{F}$. Using a recent isoperimetric inequality of Talagrand, we obtain the necessary and sufficient conditions for the large deviation estimations, the moderate deviation estimations and the LIL of $L_n(\cdot)$ in the Banach space of bounded functionals $\mathscr{l}_\infty(\mathscr{F})$. The extension to the unbounded functionals is also discussed.
Publié le : 1994-01-14
Classification:
Large deviations,
moderate deviations,
law of iterated logarithm (LIL),
isoperimetric inequality,
Smirnov-Kolmogorov theorem,
60F10,
60B12,
60G50
@article{1176988846,
author = {Wu, Liming},
title = {Large Deviations, Moderate Deviations and LIL for Empirical Processes},
journal = {Ann. Probab.},
volume = {22},
number = {4},
year = {1994},
pages = { 17-27},
language = {en},
url = {http://dml.mathdoc.fr/item/1176988846}
}
Wu, Liming. Large Deviations, Moderate Deviations and LIL for Empirical Processes. Ann. Probab., Tome 22 (1994) no. 4, pp. 17-27. http://gdmltest.u-ga.fr/item/1176988846/