A law of the iterated logarithm for distributions in the generalized domain of attraction of a nondegenerate Gaussian law

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.