Summary
When operators T n exist such that for sums S n of n i.i.d. copies of a finite-dimensional random vector X we have T n S n is shift-convergent in distribution to a standard Gaussian law, a necessary and sufficient condition on the distribution of X is given for the appropriate law of the iterated logarithm using the operators T n to hold. Our result extends certain well-known real line L.I.L.'s; it utilizes a necessary and sufficient condition due to Hahn and Klass for T n to exist giving a Gaussian limit law, and employs a second moment technique due to Kuelbs and Zinn.
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Charles Weiner, D. A law of the iterated logarithm for distributions in the generalized domain of attraction of a nondegenerate Gaussian law. Probab. Th. Rel. Fields 72, 337–357 (1986). https://doi.org/10.1007/BF00334190
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DOI: https://doi.org/10.1007/BF00334190