## The Annals of Probability

### Toward a General Law of the Iterated Logarithm in Banach Space

Uwe Einmahl

#### Abstract

A general bounded law of the iterated logarithm for Banach space valued random variables is established. Our results implies: (a) the bounded LIL of Ledoux and Talagrand, (b) a bounded LIL for random variables in the domain of attraction of a Gaussian law and (c) new LIL results for random variables outside the domain of attraction of a Gaussian law in cases where the classical norming sequence $\{\sqrt{nLLn}\}$ does not work. Basic ingredients of our proof are an infinite-dimensional Fuk-Nagaev type inequality and an infinite-dimensional version of Klass's $K$-function.

#### Article information

Source
Ann. Probab., Volume 21, Number 4 (1993), 2012-2045.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176989009

Digital Object Identifier
doi:10.1214/aop/1176989009

Mathematical Reviews number (MathSciNet)
MR1245299

Zentralblatt MATH identifier
0790.60034

JSTOR