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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Maxima and high level excursions of stationary Gaussian processes

Author: Simeon M. Berman
Journal: Trans. Amer. Math. Soc. 160 (1971), 65-85
MSC: Primary 60.50
MathSciNet review: 0290449
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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

$\displaystyle (0.1)\quad \mathop {\lim }\limits_{t \to 0} \frac{{g(ct)}}{{g(t)}} = 1,\quad for\;every\;c > 0,$

such that

$\displaystyle (0.2)\quad 1 - r(t) \sim g(\vert t\vert)\vert t{\vert^\alpha },\quad t \to 0.$

For $ u > 0$, let $ v$ be defined (in terms of $ u$) as the unique solution of

$\displaystyle (0.3)\quad {u^2}g(1/v){v^{ - \alpha }} = 1.$

Let $ {I_A}$ be the indicator of the event $ A$; then

$\displaystyle \int_0^T {{I_{[X(s) > u]}}ds} $

represents the time spent above $ u$ by $ X(s),0 \leqq s \leqq T$. It is shown that the conditional distribution of

$\displaystyle (0.4)\quad v\int_0^T {{I_{[X(s) > u]}}ds,} $

given that it is positive, converges for fixed $ T$ and $ u \to \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

$\displaystyle (0.5)\quad {F^{(p)}}\;is\;absolutely\;continuous\;for\;some\;p > 0,$

then $ \max (X(s):0 \leqq s \leqq t)$, properly normalized, has, for $ t \to \infty $, the limiting extreme value distribution $ \exp ( - {e^{ - x}})$.

If, in addition to (0.2), it is assumed that

$\displaystyle (0.6)\quad F(\lambda )\;is\; absolutely \;continuous\; with\; the\; derivative\; f(\lambda ),$


$\displaystyle (0.7)\quad \mathop {\lim }\limits_{h \to 0} \log h\int_{ - \infty }^\infty {\vert f(\lambda } + h) - f(\lambda )\vert d\lambda = 0,$

then (0.4) has, for $ u \to \infty $ and $ T \to \infty $, a limiting distribution whose Laplace-Stieltjes transform is

$\displaystyle (0.8)\quad \exp [{\text{constant}}\int_0^\infty {} ({e^{ - \lambda \xi }} - 1)d{\Psi _\alpha }(x)],\quad \lambda > 0.$

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Keywords: Conditional distribution, limiting distribution, extreme value distribution, asymptotic independence, excursion over high level, sample function maximum, occupation time, correlation function, local behavior, spectral distribution, absolute continuity, method of moments, Laplace-Stieltjes transform
Article copyright: © Copyright 1971 American Mathematical Society