## The Annals of Probability

- Ann. Probab.
- Volume 17, Number 3 (1989), 1170-1185.

### Unusual Cluster Sets for the LIL Sequence in Banach Space

#### Abstract

Let $S_n = X_1 + \cdots + X_n$, where $X_1, X_2, \cdots$ are iid Banach-space-valued random variables with weak mean 0 and weak second moments. Let $K$ be the unit ball of the reproducing kernel Hilbert space associated to the covariance of $X$. The cluster set $A$ of $\{S_n/(2n \log \log n)^{1/2}\}$ is known to be a.s. either empty or have form $\alpha K$, with $0 \leq \alpha \leq 1$ determined by a series condition. To show that this series condition is a complete characterization of $A$, examples are given to show that all $\alpha \in \lbrack 0, 1)$ do occur; $A = \phi$ and $\alpha = 1$ are already known possibilities. A regularity condition is given under which $A$ must be either $\phi$ or $K$. Under stronger moment conditions, a natural necessary and sufficient condition for $A = \phi$ is given.

#### Article information

**Source**

Ann. Probab., Volume 17, Number 3 (1989), 1170-1185.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991263

**Digital Object Identifier**

doi:10.1214/aop/1176991263

**Mathematical Reviews number (MathSciNet)**

MR1009451

**Zentralblatt MATH identifier**

0694.60005

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 60F15: Strong theorems

**Keywords**

Law of the iterated logarithm cluster set Banach-space-valued random variables

#### Citation

Alexander, Kenneth S. Unusual Cluster Sets for the LIL Sequence in Banach Space. Ann. Probab. 17 (1989), no. 3, 1170--1185. doi:10.1214/aop/1176991263. https://projecteuclid.org/euclid.aop/1176991263