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 Γ n T n \Gamma _n T_n where T n = ∑ j = 1 n Z j T_n=\sum _{j=1}^n Z_j is a sum of i.i.d. d-dimensional standard normal random vectors and Γ n \Gamma _n is a convergent sequence of symmetric non-negative definite ( d , d ) (d,d) -matrices. In the process we derive new bounds for the tail probabilities of d d -dimensional normally distributed random vectors.