## The Annals of Probability

- Ann. Probab.
- Volume 11, Number 1 (1983), 102-122.

### Some Results on the Cluster Set $C\bigg(\bigg{\frac{S_n}{a_n}\bigg}\bigg)$ and the LIL

#### Abstract

We investigate the cluster set $C(\{S_n/a_n\})$ under conditions necessary for the bounded law of the iterated logarithm, and obtain necessary and sufficient conditions for the LIL in spaces satisfying a certain comparison principle. In particular, these results settle some previously unanswered questions in the Hilbert space setting.

#### Article information

**Source**

Ann. Probab., Volume 11, Number 1 (1983), 102-122.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993662

**Digital Object Identifier**

doi:10.1214/aop/1176993662

**Mathematical Reviews number (MathSciNet)**

MR682803

**Zentralblatt MATH identifier**

0507.60002

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60B05: Probability measures on topological spaces

Secondary: 60B11: Probability theory on linear topological spaces [See also 28C20] 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 60F10: Large deviations 60F15: Strong theorems 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11] 60B10: Convergence of probability measures

**Keywords**

Law of the iterated logarithm cluster set smooth norm spaces type 2 space upper Gaussian comparison principle

#### Citation

de Acosta, A.; Kuelbs, J. Some Results on the Cluster Set $C\bigg(\bigg{\frac{S_n}{a_n}\bigg}\bigg)$ and the LIL. Ann. Probab. 11 (1983), no. 1, 102--122. doi:10.1214/aop/1176993662. https://projecteuclid.org/euclid.aop/1176993662