Open Access
January, 1994 Large Deviations, Moderate Deviations and LIL for Empirical Processes
Liming Wu
Ann. Probab. 22(1): 17-27 (January, 1994). DOI: 10.1214/aop/1176988846


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.


Download Citation

Liming Wu. "Large Deviations, Moderate Deviations and LIL for Empirical Processes." Ann. Probab. 22 (1) 17 - 27, January, 1994.


Published: January, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0793.60032
MathSciNet: MR1258864
Digital Object Identifier: 10.1214/aop/1176988846

Primary: 60F10
Secondary: 60B12 , 60G50

Keywords: Isoperimetric inequality , large deviations , law of iterated logarithm (LIL) , Moderate deviations , Smirnov-Kolmogorov theorem

Rights: Copyright © 1994 Institute of Mathematical Statistics

Vol.22 • No. 1 • January, 1994
Back to Top