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Teoriya Veroyatnostei i ee Primeneniya, 1994, Volume 39, Issue 3, Pages 641–649 (Mi tvp3840)  

This article is cited in 1 scientific paper (total in 1 paper)

Short Communications

Asymptotic properties of distributions of the maximum of a Gaussian nonstationary process occurring in covariance statistic

E. I. Ostrovskii, s. Yu. Tsykunovaa

a Obninsk Institute for Nuclear Power Engineering, The Faculty of Cybernetics
Full-text PDF (391 kB) Citations (1)
Abstract: The paper considers a separable Gaussian centered process $\eta (t)$ with a covariance function of the type
$$ \mathbf{M}\eta(t)\eta(s)=4\pi\int_{-\infty}^{+\infty}\cos\lambda t\cos\lambda sf^2(\lambda)\,d\lambda $$
for different restrictions on the spectral density $f(\lambda )$.Such processes appear as weak limits of normed deviations of empirical covariance function
$$ \eta(t)=\lim_{T\to\infty}\sqrt T(r_T(t)-r(t)) $$
as $T\to\infty$. Here $f(\lambda)=(2\pi)^{-1}\int\exp(-i\lambda t)r(t)\,dt$. The paper studies the asymptotic behavior of a probability
$$ P(u,s)=\mathbf{P}\biggl(\sup_{|t|<s}|\eta(t)|>u\biggr) $$
(as $u\to\infty$). Either the exact asymptotic or upper and lower estimates differing by a multiplicative constant are obtained for this probability. The case of Gaussian centered separable field is also considered. The results obtained may be applied for constructing the confidence interval for $r(t)$ in the uniform norm.
Keywords: covariance function, exact asymptotic, spectral density, separable centered Gaussian field, Talagrand theorem.
Received: 29.05.1990
Revised: 03.10.1991
English version:
Theory of Probability and its Applications, 1994, Volume 39, Issue 3, Pages 527–534
DOI: https://doi.org/10.1137/1139039
Bibliographic databases:
Document Type: Article
Language: Russian
Citation: E. I. Ostrovskii, s. Yu. Tsykunova, “Asymptotic properties of distributions of the maximum of a Gaussian nonstationary process occurring in covariance statistic”, Teor. Veroyatnost. i Primenen., 39:3 (1994), 641–649; Theory Probab. Appl., 39:3 (1994), 527–534
Citation in format AMSBIB
\Bibitem{OstTsy94}
\by E.~I.~Ostrovskii, s.~Yu.~Tsykunova
\paper Asymptotic properties of distributions of the maximum of a Gaussian nonstationary process occurring in covariance statistic
\jour Teor. Veroyatnost. i Primenen.
\yr 1994
\vol 39
\issue 3
\pages 641--649
\mathnet{http://mi.mathnet.ru/tvp3840}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1347192}
\zmath{https://zbmath.org/?q=an:0834.60039}
\transl
\jour Theory Probab. Appl.
\yr 1994
\vol 39
\issue 3
\pages 527--534
\crossref{https://doi.org/10.1137/1139039}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1995TF06800014}
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  • https://www.mathnet.ru/eng/tvp/v39/i3/p641
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Теория вероятностей и ее применения Theory of Probability and its Applications
     
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