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 Probl. Peredachi Inf., 2010, Volume 46, Issue 1, Pages 68–93 (Mi ppi2010)

Large Systems

Small deviations for two classes of Gaussian stationary processes and $L^p$-functionals, $0<p\le\infty$

V. R. Fatalov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: Let $w(t)$ be a standard Wiener process, $w(0)=0$, and let $\eta_a(t)=w(t+a)-w(t)$, $t\ge0$, be increments of the Wiener process, $a>0$. Let $Z_a(t)$, $t\in[0,2a]$, be a zeromean Gaussian stationary a.s. continuous process with a covariance function of the form $\mathbf EZ_a(t)Z_a(s)=\frac12[a-|t-s|]$, $t,s\in[0,2a]$. For $0<p<\infty$, we prove results on sharp asymptotics as $\varepsilon\to0$ of the probabilities
$$\mathbf P\{\int_0^T|\eta_a(t)|^p dt\le\varepsilon^p\}\quadäëÿ T\le a,\qquad\mathbf P\{\int_0^T|Z_a(t)|^p dt\le\varepsilon^p\}\quadäëÿ T<2a,$$
and compute similar asymptotics for the sup-norm. Derivation of the results is based on the method of comparing with a Wiener process. We present numerical values of the asymptotics in the case $p=1$, $p=2$, and for the sup-norm. We also consider application of the obtained results to one functional quantization problem of information theory.

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English version:
Problems of Information Transmission, 2010, 46:1, 62–85

Bibliographic databases:

UDC: 621.391.1+519.21
Revised: 17.11.2009

Citation: V. R. Fatalov, “Small deviations for two classes of Gaussian stationary processes and $L^p$-functionals, $0<p\le\infty$”, Probl. Peredachi Inf., 46:1 (2010), 68–93; Problems Inform. Transmission, 46:1 (2010), 62–85

Citation in format AMSBIB
\Bibitem{Fat10} \by V.~R.~Fatalov \paper Small deviations for two classes of Gaussian stationary processes and $L^p$-functionals, $0<p\le\infty$ \jour Probl. Peredachi Inf. \yr 2010 \vol 46 \issue 1 \pages 68--93 \mathnet{http://mi.mathnet.ru/ppi2010} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2675299} \transl \jour Problems Inform. Transmission \yr 2010 \vol 46 \issue 1 \pages 62--85 \crossref{https://doi.org/10.1134/S0032946010010060} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000276978000006} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77951530398} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. R. Fatalov, “Laplace-type exact asymptotic formulas for the Bogoliubov Gaussian measure”, Theoret. and Math. Phys., 168:2 (2011), 1112–1149
2. V. R. Fatalov, “Negative-order moments for $L^p$-functionals of Wiener processes: exact asymptotics”, Izv. Math., 76:3 (2012), 626–646
3. V. R. Fatalov, “Asymptotic behavior of small deviations for Bogoliubov's Gaussian measure in the $L^p$ norm, $2\le p\le\infty$”, Theoret. and Math. Phys., 173:3 (2012), 1720–1733
4. V. R. Fatalov, “Ergodic means for large values of $T$ and exact asymptotics of small deviations for a multi-dimensional Wiener process”, Izv. Math., 77:6 (2013), 1224–1259
5. V. R. Fatalov, “Gaussian Ornstein–Uhlenbeck and Bogoliubov processes: asymptotics of small deviations for $L^p$-functionals, $0<p<\infty$”, Problems Inform. Transmission, 50:4 (2014), 371–389
6. Kirichenko A.A., Nikitin Ya.Yu., “Precise Small Deviations in l-2 of Some Gaussian Processes Appearing in the Regression Context”, Cent. Eur. J. Math., 12:11 (2014), 1674–1686
7. V. R. Fatalov, “Weighted $L^p$, $p\ge2$, for a wiener process: Exact asymptoties of small deviations”, Moscow University Mathematics Bulletin, 70:2 (2015), 68–73
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