## The Annals of Probability

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

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

#### 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**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)

**Keywords**

Bounded law of the iterated logarithm $K$-function LIL behavior randomization Rademacher random variables

#### Citation

Einmahl, Uwe. Toward a General Law of the Iterated Logarithm in Banach Space. Ann. Probab. 21 (1993), no. 4, 2012--2045. doi:10.1214/aop/1176989009. https://projecteuclid.org/euclid.aop/1176989009