## The Annals of Probability

- Ann. Probab.
- Volume 17, Number 2 (1989), 737-759.

### Characterization of the Cluster Set of the LIL Sequence in Banach Space

#### Abstract

Let $S_n = X_1 + \cdots + X_n$, where $X_1, X_2, \ldots$ 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 $X$. We show that the cluster set of $\{S_n/(2n \log \log n)^{1/2}\}$ either is empty or has the form $\alpha K$, where $0 \leq \alpha \leq 1$. A series condition is given which determines the value of $\alpha$. In a companion paper, examples are given to show that all $\alpha \in \lbrack 0, 1 \rbrack$ do occur.

#### Article information

**Source**

Ann. Probab., Volume 17, Number 2 (1989), 737-759.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991424

**Digital Object Identifier**

doi:10.1214/aop/1176991424

**Mathematical Reviews number (MathSciNet)**

MR985387

**Zentralblatt MATH identifier**

0679.60006

**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. Characterization of the Cluster Set of the LIL Sequence in Banach Space. Ann. Probab. 17 (1989), no. 2, 737--759. doi:10.1214/aop/1176991424. https://projecteuclid.org/euclid.aop/1176991424