The Annals of Probability

Sojourns and Extremes of Gaussian Processes

Simeon M. Berman

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Abstract

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.

Article information

Source
Ann. Probab., Volume 2, Number 6 (1974), 999-1026.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996495

Digital Object Identifier
doi:10.1214/aop/1176996495

Mathematical Reviews number (MathSciNet)
MR372976

Zentralblatt MATH identifier
0298.60026

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

Citation

Berman, Simeon M. Sojourns and Extremes of Gaussian Processes. Ann. Probab. 2 (1974), no. 6, 999--1026. doi:10.1214/aop/1176996495. https://projecteuclid.org/euclid.aop/1176996495


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Corrections

  • See Correction: Simeon M. Berman. Correction: Correction to "Sojourns and Extremes of Gaussian Processes". Ann. Probab., Volume 12, Number 1 (1984), 281--281.
  • See Correction: Simeon M. Berman. Correction Notes: Correction to "Sojourns and Extremes of Gaussian Processes". Ann. Probab., Volume 8, Number 5 (1980), 999--999.