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June, 1973 Excursions of Stationary Gaussian Processes above High Moving Barriers
Simeon M. Berman
Ann. Probab. 1(3): 365-387 (June, 1973). DOI: 10.1214/aop/1176996932


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.


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Simeon M. Berman. "Excursions of Stationary Gaussian Processes above High Moving Barriers." Ann. Probab. 1 (3) 365 - 387, June, 1973.


Published: June, 1973
First available in Project Euclid: 19 April 2007

zbMATH: 0259.60015
MathSciNet: MR388514
Digital Object Identifier: 10.1214/aop/1176996932

Primary: 60G10
Secondary: 60F99 , 60G15 , 60G17

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

Rights: Copyright © 1973 Institute of Mathematical Statistics

Vol.1 • No. 3 • June, 1973
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