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 Mat. Sb., 2003, Volume 194, Number 3, Pages 61–82 (Mi msb721)

This article is cited in 8 scientific papers (total in 8 papers)

Asymptotics of large deviations of Gaussian processes of Wiener type for $L^p$-functionals, $p>0$, and the hypergeometric function

V. R. Fatalov

M. V. Lomonosov Moscow State University

Abstract: A general result is obtained on exact asymptotics of the probabilities
$$\mathsf P\{\int_0^1|\xi(t)|^p dt>u^p\}$$
as $u\to\infty$ and $p>0$ for Gaussian processes $\xi(t)$.
The general theorem is applied for the calculation of these asymptotics in the cases of the following processes: the Wiener process $w(t)$, the Brownian bridge, and the stationary Gaussian process $\eta(t):=w(t+1)-w(t)$, $t\in\mathbb R^1$.
The Laplace method in Banach spaces is used. The calculations of the constants reduce to solving an extremum problem for the action functional and studying the spectrum of a differential operator of the second order of Sturm–Liouville type.

DOI: https://doi.org/10.4213/sm721

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English version:
Sbornik: Mathematics, 2003, 194:3, 369–390

Bibliographic databases:

UDC: 519.2
MSC: Primary 60F10; Secondary 60G10, 60G15, 60J65
Received: 23.05.2002

Citation: V. R. Fatalov, “Asymptotics of large deviations of Gaussian processes of Wiener type for $L^p$-functionals, $p>0$, and the hypergeometric function”, Mat. Sb., 194:3 (2003), 61–82; Sb. Math., 194:3 (2003), 369–390

Citation in format AMSBIB
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This publication is cited in the following articles:
1. V. R. Fatalov, “Large deviations for Gaussian processes in Hölder norm”, Izv. Math., 67:5 (2003), 1061–1079
2. V. R. Fatalov, “The Laplace method for small deviations of Gaussian processes of Wiener type”, Sb. Math., 196:4 (2005), 595–620
3. Fatalov V.P., “Letter to the Editors”, Theory Probab. Appl., 51:3 (2007), 561–563
4. V. R. Fatalov, “Exact Asymptotics of Large Deviations of Stationary Ornstein–Uhlenbeck Processes for $L^p$-Functional, $p>0$”, Problems Inform. Transmission, 42:1 (2006), 46–63
5. V. R. Fatalov, “Exact Asymptotics of Distributions of Integral Functionals of the Geometric Brownian Motion and Other Related Formulas”, Problems Inform. Transmission, 43:3 (2007), 233–254
6. Louchard G., Janson S., “Tail estimates for the Brownian excursion area and other Brownian areas”, Electron. J. Probab., 12:58 (2007), 1600–1632
7. V. R. Fatalov, “Exact asymptotics of Laplace-type Wiener integrals for $L^p$-functionals”, Izv. Math., 74:1 (2010), 189–216
8. FuChang Gao, XiangFeng Yang, “Upper tail probabilities of integrated Brownian motions”, Sci. China Math, 2015
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