The Law of the Iterated Logarithm for Gaussian Processes

Abstract

Strassen's law of the iterated logarithm for Brownian motion is extended to a class of Gaussian processes. Let $\{X(t), t \geqq 0\}$ be a real continuous Gaussian process with $X(0) = 0$, mean zero and continuous covariance kernel $R(s, t)$. Define a random sequence $\{f_n(t, \omega)\}$ in $C\lbrack 0, 1 \rbrack$ by $f_n(t, \omega) = X(nt, \omega)/(2R(n, n) \log \log n)^{\frac{1}{2}}$. Under certain conditions on $R$ it is shown that with probability one $\{f_n(t, \omega)\}$ is equicontinuous and the set of its limit points is the unit ball of the reproducing kernel Hilbert space with reproducing kernel $\Gamma$ determined by $R$. The result generalizes the author's earlier result (1972).