## The Annals of Probability

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

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

V. Goodman, J. Kuelbs, and J. Zinn

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60B05: Probability measures on topological spaces

Secondary: 60F05: Central limit and other weak theorems 60F10: Large deviations 60F15: Strong theorems 28A40 60B10: Convergence of probability measures

**Keywords**

Law of the iterated logarithm cluster set central limit theorem

#### Citation

Goodman, V.; Kuelbs, J.; Zinn, J. Some Results on the LIL in Banach Space with Applications to Weighted Empirical Processes. Ann. Probab. 9 (1981), no. 5, 713--752. doi:10.1214/aop/1176994305. https://projecteuclid.org/euclid.aop/1176994305