Asymptotic Properties of Gaussian Processes

Abstract

We study separable mean zero Gaussian processes $X(t)$ with correlation $\rho (t, s)$ for which $1 - \rho (t, s)$ is asymptotic to a regularly varying (at zero) function of $|t - s|$ with exponent $0 < \alpha \leqq 2$. For such processes, we obtain the asymptotic distribution of the maximum of $X(t)$. This result is used to obtain a result for $X(t)$ as $t \rightarrow \infty$ similar to the so-called law of the iterated logarithm.