## The Annals of Probability

### Some Results on the LIL in Banach Space with Applications to Weighted Empirical Processes

#### Abstract

We examine the cluster set of $S_n/a_n$ for Banach space valued random variables, and investigate the relationship between the central limit theorem and the law of the iterated logarithm in this setting. In the case of Hilbert space valued random variables, necessary and sufficient conditions are given for the law of the iterated logarithm. Some interesting examples are also included. We then apply our results to weighted empiricals both in the supremum norm and the $L^2\lbrack 0, 1\rbrack$ norm.

#### Article information

Source
Ann. Probab., Volume 9, Number 5 (1981), 713-752.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176994305

Digital Object Identifier
doi:10.1214/aop/1176994305

Mathematical Reviews number (MathSciNet)
MR628870

Zentralblatt MATH identifier
0472.60004

JSTOR