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February, 1982 Sojourns and Extremes of Stationary Processes
Simeon M. Berman
Ann. Probab. 10(1): 1-46 (February, 1982). DOI: 10.1214/aop/1176993912


Let $X(t), -\infty < t < \infty$, be a real stationary stochastic process with continuous sample functions. For $t > 0$, put $L_t(u) =$ Lebesgue measure of $\{s: 0 \leq s \leq t, X(s) > u\}$ and $M(t) = \max(X(s): 0 \leq s \leq t)$. For several years the author has studied the limiting properties of these random variables in the case where $X(t)$ is a Gaussian process and under two kinds of limiting operations: i) $t$ fixed and $u \rightarrow \infty$; ii) $t \rightarrow \infty$ and $u = u(t) \rightarrow \infty$ as a function of $t$. The purpose of this paper is to show how the methods developed in the Gaussian case can be extended to the general, not necessarily Gaussian case. This is illustrated by applications of some of the results to specific examples of non-Gaussian processes, and classes of processes containing a Gaussian subclass.


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Simeon M. Berman. "Sojourns and Extremes of Stationary Processes." Ann. Probab. 10 (1) 1 - 46, February, 1982.


Published: February, 1982
First available in Project Euclid: 19 April 2007

zbMATH: 0498.60035
MathSciNet: MR637375
Digital Object Identifier: 10.1214/aop/1176993912

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

Keywords: Extreme values , Gaussian process , sample function maximum , Sojourn , stationary process

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 1 • February, 1982
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