## The Annals of Probability

- Ann. Probab.
- Volume 13, Number 3 (1985), 825-859.

### The LIL when $X$ is in the Domain of Attraction of a Gaussian Law

#### Abstract

If $X$ takes values in a Banach space $B$ and is in the domain of normal attraction of a Gaussian law on $B$ with $EX = 0, E(\|X\|^2/L_2\|X\|) < \infty$, then it is known that $X$ satisfies the compact law of the iterated logarithm as described in Goodman, Kuelbs and Zinn [9], Theorem 4.1. In this paper the analogous result is demonstrated when $X$ is merely in the domain of attraction of a Gaussian law. The functional LIL is also obtained in this setting. These results refine Corollary 7 of Kuelbs and Zinn [22], as well as various functional LILs.

#### Article information

**Source**

Ann. Probab., Volume 13, Number 3 (1985), 825-859.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176992910

**Digital Object Identifier**

doi:10.1214/aop/1176992910

**Mathematical Reviews number (MathSciNet)**

MR799424

**Zentralblatt MATH identifier**

0572.60010

**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 domain of attraction of a Gaussian random variable

#### Citation

Kuelbs, J. The LIL when $X$ is in the Domain of Attraction of a Gaussian Law. Ann. Probab. 13 (1985), no. 3, 825--859. doi:10.1214/aop/1176992910. https://projecteuclid.org/euclid.aop/1176992910