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December, 1974 Sojourns and Extremes of Gaussian Processes
Simeon M. Berman
Ann. Probab. 2(6): 999-1026 (December, 1974). DOI: 10.1214/aop/1176996495


Let $X(t), 0 \leqq t \leqq 1$, be a real Gaussian process with mean 0 and continuous sample functions. For $u > 0$, form the process $u(X(t) - u)$. In this paper two related problems are studied. (i) Let $G$ be a nonnegative measurable function, and put $L = \int^1_0 G(u(X(t) - u)) dt$. For certain classes of processes $X$ and functions $G$, we find, for $u \rightarrow \infty$, the limiting conditional distribution of $L$ given that it is positive. (ii) For the same class of processes $X$, we find the asymptotic form of $P(\max_{\lbrack 0,1 \rbrack} X(t) > u)$ for $u \rightarrow \infty$. Finally, these results are extended to the process with the "moving barrier," $X(t) - f(t)$, where $f$ is a continuous function.


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Simeon M. Berman. "Sojourns and Extremes of Gaussian Processes." Ann. Probab. 2 (6) 999 - 1026, December, 1974.


Published: December, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0298.60026
MathSciNet: MR372976
Digital Object Identifier: 10.1214/aop/1176996495

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

Keywords: Gaussian proces , level barrier , local stationarity , moving barrier Sample function miximum , regular variation , weak compactness

Rights: Copyright © 1974 Institute of Mathematical Statistics


Vol.2 • No. 6 • December, 1974
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