Feller's upper-lower class test in Euclidean space

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 $Γ_n T_n$ where $T_n=\sum_{j=1}^n Z_j$ is a sum of i.i.d. d-dimensional standard normal random vectors and $Γ_n$ 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.

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