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.
Publié le : 1973-06-14
Classification:
Stationary Gaussian process,
moving barrier,
regular variation,
integral equation,
conditional distribution,
excursion over barrier,
sample function maximum,
weak convergence,
60G10,
60G15,
60G17,
60F99
@article{1176996932,
author = {Berman, Simeon M.},
title = {Excursions of Stationary Gaussian Processes above High Moving Barriers},
journal = {Ann. Probab.},
volume = {1},
number = {5},
year = {1973},
pages = { 365-387},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996932}
}
Berman, Simeon M. Excursions of Stationary Gaussian Processes above High Moving Barriers. Ann. Probab., Tome 1 (1973) no. 5, pp. 365-387. http://gdmltest.u-ga.fr/item/1176996932/