The Annals of Probability

Excursions of Stationary Gaussian Processes above High Moving Barriers

Simeon M. Berman

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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

Subjects
Primary: 60G10: Stationary processes
Secondary: 60G15: Gaussian processes 60G17: Sample path properties 60F99: None of the above, but in this section

Keywords
Stationary Gaussian process moving barrier regular variation integral equation conditional distribution excursion over barrier sample function maximum weak convergence

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


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