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This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.Maxima and High Level Excursions of Stationary Gaussian Processes
Simeon M. Berman
Transactions of the American Mathematical Society
Vol. 160 (Oct., 1971), pp. 6585
Published by: American Mathematical Society
DOI: 10.2307/1995791
Stable URL: http://www.jstor.org/stable/1995791
Page Count: 21
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Abstract
Let $X(t), t \geqq 0$, be a stationary Gaussian process with mean $0$, variance $1$ and covariance function $r(t)$. The sample functions are assumed to be continuous on every interval. Let $r(t)$ be continuous and nonperiodic. Suppose that there exists $\alpha, 0 < \alpha \leqq 2$, and a continuous, increasing function $g(t), t \geqq 0$, satisfying $$(0.1) \lim {t \rightarrow 0} {\frac {g(ct)} {g(t)} = 1, \text{for every} c > 0$$, such that $$(0.2) 1  r(t) \sim g(t)t^\alpha, t \rightarrow 0$$. For $u > 0$, let $\nu$ be defined (in terms of $u$) as the unique solution of $$(0.3) u^2 g(1/\nu)\nu ^{\alpha} = 1$$. Let $I_A$ be the indicator of the event $A$; then $$\int^T_0 I_{\lbrack X(s) > u\ rbrack} ds$$ represents the time spent above $u$ by $X(s), 0 \leqq s \leqq T$. It is shown that the conditional distribution of $$(0.4) \nu \int^T_0 I_{\lbrack X(s) > u \rbrack} ds$$, given that it is positive, converges for fixed $T$ and $u \rightarrow \infty$ to a limiting distribution $\Psi_\alpha$, which depends only on $\alpha$ but not on $T$ or $g$. Let $F(\lambda)$ be the spectral distribution function corresponding to $r(t)$. Let $F^{(p)}(\lambda)$ be the iterated $p$fold convolution of $F(\lambda)$. If, in addition to (0.2), it is assumed that $$(0.5) F^{(p)} is absolutely continuous for some p > 0$$, then max $(X(s): 0 \leqq s \leqq t)$, properly normalized, has, for $t \rightarrow \infty$, the limiting extreme value distribution exp $(e^{x})$. If, in addition to (0.2), it is assumed that $$(0.6) F(\lambda) \text{is absolutely continuous with the derivative} f(\lambda)$$, and $$(0.7) \lim _{h \rightarrow 0} \log h \int^\infty_{\infty} f(\lambda + h)  f(\lambda) d\lambda = 0$$, then (0.4) has, for $u \rightarrow \infty$, and $T \rightarrow \infty$, a limiting distribution whose LaplaceStieltjes transform is $$(0.8) \exp \big\lbrack constant \int^\infty_0 (e ^{\lambda x}1) d \Psi \alpha (x) \big\rbrack, \lambda > 0.$$
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Transactions of the American Mathematical Society © 1971 American Mathematical Society