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Uspekhi Mat. Nauk, 2003, Volume 58, Issue 4(352), Pages 89–134 (Mi umn643)  

This article is cited in 18 scientific papers (total in 18 papers)

Constants in the asymptotics of small deviation probabilities for Gaussian processes and fields

V. R. Fatalov

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

Abstract: This paper presents a survey of results on computing the small deviation asymptotics for Gaussian measures, that is, the asymptotics of the probabilities
$$ \mu(\varepsilon D), \qquad \varepsilon\to0, $$
where $D$ is a bounded domain in a Banach space $(B,{\|\cdot\|})$ (for example, $D=\{x\in B:\|x\|\leqslant 1\}$) and $\mu$ a Gaussian measure on $B$.
The main attention is focused on calculating the values of constants in the exact or logarithmic asymptotics. The survey contains new numerical results; some erroneous assertions in previous papers on this topic are also noted.
The following classes of Gaussian processes and fields are studied in detail: Wiener processes and related processes, Brownian bridges, Bessel processes, vector Wiener processes, Gaussian Markov processes, Gaussian processes with stationary increments, fractional Ornstein–Uhlenbeck processes, $n$-parameter fractional Brownian motion, $n$-parameter Wiener–Chentsov fields, and the Wiener pillow. Results on small deviations are presented in diverse norms, namely, the sup-norm, Hilbert norms, $L^p$-norms, Hölder norms, Orlicz norms, and weighted sup-norms.
About 30 problems concerned with finding exact constants in asymptotic expressions for small deviations are posed.
The relation to Chung's law of the iterated logarithm is also considered, and a number of other results are presented.

DOI: https://doi.org/10.4213/rm643

Full text: PDF file (570 kB)
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English version:
Russian Mathematical Surveys, 2003, 58:4, 725–772

Bibliographic databases:

UDC: 519.21
MSC: Primary 60G15, 60B11; Secondary 60J65, 60J25, 60F10, 60F15, 28C20, 46T12
Received: 27.11.2001

Citation: V. R. Fatalov, “Constants in the asymptotics of small deviation probabilities for Gaussian processes and fields”, Uspekhi Mat. Nauk, 58:4(352) (2003), 89–134; Russian Math. Surveys, 58:4 (2003), 725–772

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. R. Fatalov, “Asymptotics of large deviations of Gaussian processes of Wiener type for $L^p$-functionals, $p>0$, and the hypergeometric function”, Sb. Math., 194:3 (2003), 369–390  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. V. R. Fatalov, “Large deviations for Gaussian processes in Hölder norm”, Izv. Math., 67:5 (2003), 1061–1079  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. A. I. Nazarov, Ya. Yu. Nikitin, “Logarithmic $L_2$-small ball asymptotics for some fractional Gaussian processes”, Theory Probab. Appl., 49:4 (2005), 645–658  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. A. I. Nazarov, “Logarithmic $L_2$-small ball asymptotics with respect to self-similar measure for some Gaussian processes”, J. Math. Sci. (N. Y.), 133:3 (2006), 1314–1327  mathnet  crossref  mathscinet  zmath  elib
    5. V. R. Fatalov, “The Laplace method for small deviations of Gaussian processes of Wiener type”, Sb. Math., 196:4 (2005), 595–620  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. 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  mathnet  crossref  mathscinet  zmath  elib  elib
    7. V. R. Fatalov, “Occupation times and exact asymptotics of small deviations of Bessel processes for $L^p$-norms with $p>0$”, Izv. Math., 71:4 (2007), 721–752  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    8. V. R. Fatalov, “Exact Asymptotics of Small Deviations for a Stationary Ornstein–Uhlenbeck Process and Some Gaussian Diffusion Processes in the $L_p$-Norm, $2\le p\le\infty$”, Problems Inform. Transmission, 44:2 (2008), 138–155  mathnet  crossref  mathscinet  isi  elib
    9. V. R. Fatalov, “Exact asymptotics of Laplace-type Wiener integrals for $L^p$-functionals”, Izv. Math., 74:1 (2010), 189–216  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    10. V. R. Fatalov, “Small deviations for two classes of Gaussian stationary processes and $L^p$-functionals, $0<p\le\infty$”, Problems Inform. Transmission, 46:1 (2010), 62–85  mathnet  crossref  mathscinet  isi
    11. V. R. Fatalov, “Negative-order moments for $L^p$-functionals of Wiener processes: exact asymptotics”, Izv. Math., 76:3 (2012), 626–646  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    12. 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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    13. Ya. Yu. Nikitin, R. S. Pusev, “The exact asymptotic of small deviations for a series of Brownian functionals”, Theory Probab. Appl., 57:1 (2013), 60–81  mathnet  crossref  crossref  zmath  isi  elib  elib
    14. A.N.. Frolov, “Small deviations of iterated processes in the space of trajectories”, centr.eur.j.math, 11:12 (2013), 2089  crossref  mathscinet  zmath  isi
    15. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    16. 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  mathnet  crossref  mathscinet
    17. Ai X., “A Distributional Identity For the Bivariate Brownian Bridge: a Nontensor Gaussian Field”, Math. Probl. Eng., 2018, 9687039  crossref  mathscinet  isi  scopus  scopus
    18. MacKay A., Melnikov A., Mishura Yu., “Optimization of Small Deviation For Mixed Fractional Brownian Motion With Trend”, Stochastics, 90:7 (2018), 1087–1110  crossref  mathscinet  isi  scopus  scopus
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