## The Annals of Probability

- Ann. Probab.
- Volume 16, Number 3 (1988), 1242-1264.

### Characterization of the Law of the Iterated Logarithm in Banach Spaces

#### Abstract

Using a Gaussian randomization technique, we prove that a random variable $X$ with values in a Banach space $B$ satisfies the (compact) law of the iterated logarithm if and only if (i) $E(\|X\|^2/LL\|X\|) < \infty$, (ii) $\{|\langle x^\ast, X \rangle |^2; x^\ast \in B^\ast, \|x^\ast\| \leq 1\}$ is uniformly integrable and (iii) $S_n(x)/a_n\rightarrow 0$ in probability. In particular, if $B$ is of type 2, in order that $X$ satisfy the law of the iterated logarithm it is necessary and sufficient that $X$ have mean zero and satisfy (i) and (ii). The proof uses tools of the theory of Gaussian random vectors as well as by now classical arguments of probability in Banach spaces. It also sheds some light on the usual law of the iterated logarithm on the line.

#### Article information

**Source**

Ann. Probab., Volume 16, Number 3 (1988), 1242-1264.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991688

**Digital Object Identifier**

doi:10.1214/aop/1176991688

**Mathematical Reviews number (MathSciNet)**

MR942766

**Zentralblatt MATH identifier**

0662.60008

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 60B11: Probability theory on linear topological spaces [See also 28C20] 60G15: Gaussian processes 46B20: Geometry and structure of normed linear spaces

**Keywords**

Law of the iterated logarithm Banach spaces Gaussian randomization type 2

#### Citation

Ledoux, M.; Talagrand, M. Characterization of the Law of the Iterated Logarithm in Banach Spaces. Ann. Probab. 16 (1988), no. 3, 1242--1264. doi:10.1214/aop/1176991688. https://projecteuclid.org/euclid.aop/1176991688