High Level Occupation Times for Continuous Gaussian Processes

Abstract

Let $\{y(\tau), 0 \leqq \tau \leqq 1\}$ be a sample continuous Gaussian process, and let $T\lbrack y, \alpha \rbrack$ denote the time that $y(\cdot)$ spends above level $\alpha:$ $$T\lbrack y, \alpha \rbrack = \int^1_0 V(y(\tau) - \alpha) d\tau,$$ where $V(x) = 0$ or 1 according as $x \leqq 0$ or $x > 0.$ In this paper it is proved that, as $\alpha \rightarrow \infty,$ $$P\{T\lbrack y, \alpha \rbrack > \beta\} = \exp \{-(\alpha^2/2)\mathbf{k}_\beta(1 + o(1))\}$$ where $k_\beta$ is a particular functional of the covariance function of the process.