## The Annals of Probability

### Excursions of Stationary Gaussian Processes above High Moving Barriers

Simeon M. Berman

#### Abstract

Let $X(t)$ be a real stationary Gaussian process with covariance function $r(t);$ and let $f(t), t \geqq 0,$ be a nonnegative continuous function which vanishes only at $t = 0.$ Under certain conditions on $r(t)$ and $f(t),$ we find, for fixed $T > 0$ and for $u \rightarrow \infty$ (i) the asymptotic form of the probability that $X(t)$ exceeds $u + f(t)$ for some $t \in \lbrack 0, T \rbrack;$ and (ii) the conditional limiting distribution of the time spent by $X(t)$ above $u + f(t), 0 \leqq t \leqq T,$ given that the time is positive.

#### Article information

Source
Ann. Probab., Volume 1, Number 3 (1973), 365-387.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996932

Digital Object Identifier
doi:10.1214/aop/1176996932

Mathematical Reviews number (MathSciNet)
MR388514

Zentralblatt MATH identifier
0259.60015

JSTOR
links.jstor.org

#### Citation

Berman, Simeon M. Excursions of Stationary Gaussian Processes above High Moving Barriers. Ann. Probab. 1 (1973), no. 3, 365--387. doi:10.1214/aop/1176996932. https://projecteuclid.org/euclid.aop/1176996932