The Law of the Iterated Logarithm and Upper-Lower Class Tests for Partial Sums of Stationary Gaussian Sequences

Abstract

Herein, laws of the iterated logarithm and various upper-lower class refinements are established for partial sums of stationary Gaussian random variables. These stationary Gaussian random variables are not necessarily in any sense weakly dependent. For example, if the random variables are nonnegatively correlated, then the upper half of the law of the iterated logarithm holds. Under more restrictive, but still quite general hypotheses, an upper-lower class test which classifies all monotone sequences $\{\phi(n)\}$ is established.