The Law of the Iterated Logarithm over a Stationary Gaussian Sequence of Random Vectors

Abstract

Let {Xj}j = 1∞ be a stationary Gaussian sequence of random vectors with mean zero. We give sufficient conditions for the compact law of the iterate logarithm of $$(n{\text{ 2 log log }}n{\text{)}}^{{\text{ - 1/2}}} \sum\limits_{j{\text{ }} = {\text{ }}1}^n {(G(X_j ) - E[{\text{ }}G} (X_j )])$$ where G is a real function defined on ℝd with finite second moment. Our result builds on Ho,(6) who proved an upper-half of the law of iterated logarithm for a sequence of random variables.