Feller's upper-lower class test in Euclidean space

TL;DR

This paper extends Feller's upper-lower class test to the Law of the Iterated Logarithm (LIL) in Euclidean space, providing new bounds for tail probabilities of multivariate normal vectors.

Contribution

It introduces a general upper-lower class test for sums of i.i.d. multivariate normal vectors with matrix transformations, extending classical results to higher dimensions.

Findings

Derived new bounds for tail probabilities of multivariate normal vectors.

Extended Feller's test to the Euclidean space LIL.

Provided a corollary for the Hartman-Wintner LIL in multiple dimensions.

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