A law of the iterated logarithm for distributions in the generalized domain of attraction of a nondegenerate Gaussian law
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 \(T_ nS_ n\) 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 \textit{M. G. Hahn} and \textit{M. J. Klass} [Probability theory on vector spaces II, Proc., Blazejewko/Poland 1979, Lect. Notes Math. 828, 52-81 (1980; Zbl 0457.60007)] for \(T_ n\) to exist giving a Gaussian limit law, and employs a second moment technique due to \textit{J. Kuelbs} and \textit{J. Zinn} [see Ann. Probab. 11, 506-557 (1983; Zbl 0518.60009)].
scientific article; zbMATH DE number 3913335