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Journal Article

Maxima and High Level Excursions of Stationary Gaussian Processes

Simeon M. Berman
Transactions of the American Mathematical Society
Vol. 160 (Oct., 1971), pp. 65-85
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 Laplace-Stieltjes 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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