Sojourns and Extremes of Gaussian Processes

Abstract

Let $X(t), 0 \leqq t \leqq 1$, be a real Gaussian process with mean 0 and continuous sample functions. For $u > 0$, form the process $u(X(t) - u)$. In this paper two related problems are studied. (i) Let $G$ be a nonnegative measurable function, and put $L = \int^1_0 G(u(X(t) - u)) dt$. For certain classes of processes $X$ and functions $G$, we find, for $u \rightarrow \infty$, the limiting conditional distribution of $L$ given that it is positive. (ii) For the same class of processes $X$, we find the asymptotic form of $P(\max_{\lbrack 0,1 \rbrack} X(t) > u)$ for $u \rightarrow \infty$. Finally, these results are extended to the process with the "moving barrier," $X(t) - f(t)$, where $f$ is a continuous function.