Feller's upper-lower class test in Euclidean space

Abstract

We provide an extension of Feller's upper-lower class test for the Hartman-Wintner LIL to the LIL in Euclidean space. We obtain this result as a corollary to a general upper-lower class test for ΓnTn where Tn = n j=1 Zj is a sum of i.i.d. d-dimensional standard normal random vectors and Γn is a convergent sequence of symmetric non-negative definite (d, d)-matrices. In the process we derive new bounds for the tail probabilities of d-dimensional normally distributed random vectors.