Sequential Compactness of Certain Sequences of Gaussian Random Variables with Values in $C\lbrack 0, 1 \rbrack$

Abstract

Let $(X_n(t): t \in \lbrack 0, 1 \rbrack, n \geqq 1)$ be a sequence of Gaussian processes with mean zero and continuous paths on [0, 1] a.s. Let $R_n(t, s) = EX_n(t)X_n(s)$ and suppose that $(R_n: n \geqq 1)$ is uniformly convergent, on the unit square, to a covariance function $R$. It is shown in this paper that under certain conditions the normalized sequence $(Y_n(t): t \in \lbrack 0, 1 \rbrack, n \geqq 2)$ where $Y_n(t) = (2\lg n)^{-\frac{1}{2}}X_n(t)$ is, with probability one, a sequentially compact subset of $C\lbrack 0, 1 \rbrack$ and its set of limit points coincides a.s. with the unit ball in the reproducing kernel Hilbert space generated by $R$. This is Strassen's form of the iterated logarithm in its intrinsic formulation and includes a special case studied by Oodaira in a recent paper.