The Annals of Probability

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

J. Kuelbs

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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

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

First available in Project Euclid: 19 April 2007

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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

Law of the iterated logarithm cluster set domain of attraction of a Gaussian random variable


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.

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