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Teor. Veroyatnost. i Primenen., 2004, Volume 49, Issue 4, Pages 695–711 (Mi tvp189)  

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

Logarithmic $L_2$-small ball asymptotics for some fractional Gaussian processes

A. I. Nazarov, Ya. Yu. Nikitin

St. Petersburg State University, Department of Mathematics and Mechanics

Abstract: We find the logarithmic $L_2$-small ball asymptotics of some Gaussian processes related to the fractional Brownian motion (fBm), fractional Ornstein–Uhlenbeck process (fOU), and their integrated analogues. We consider also the multiparameter generalizations.

Keywords: fractional Brownian motion, fractional Ornstein–Uhlenbeck process, Lévy's Brownian motion, small deviations, spectral asymptotics.

DOI: https://doi.org/10.4213/tvp189

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English version:
Theory of Probability and its Applications, 2005, 49:4, 645–658

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Received: 17.05.2004

Citation: A. I. Nazarov, Ya. Yu. Nikitin, “Logarithmic $L_2$-small ball asymptotics for some fractional Gaussian processes”, Teor. Veroyatnost. i Primenen., 49:4 (2004), 695–711; Theory Probab. Appl., 49:4 (2005), 645–658

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. Lifshits M., Linde W., Shi Zhan, “Small deviations of Gaussian random fields in $L_q$-spaces”, Electron. J. Probab., 11 (2006), 1204–1233  crossref  mathscinet  zmath  isi  elib  scopus
    2. Neuenkirch A., “Optimal approximation of SDEs with additive fractional noise”, J. Complexity, 22:4 (2006), 459–474  crossref  mathscinet  zmath  isi  elib  scopus
    3. 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
    4. Ayache A., Linde W., “Approximation of Gaussian random fields: General results and optimal wavelet representation of the Lévy fractional motion”, J. Theoret. Probab., 21:1 (2008), 69–96  crossref  mathscinet  zmath  isi  elib  scopus
    5. Karol' A., Nazarov A., Nikitin Ya., “Small ball probabilities for Gaussian random fields and tensor products of compact operators”, Trans. Amer. Math. Soc., 360:3 (2008), 1443–1474  crossref  mathscinet  zmath  isi  elib  scopus
    6. Nazarov A., “Log-level comparison principle for small ball probabilities”, Statist. Probab. Lett., 79:4 (2009), 481–486  crossref  mathscinet  zmath  isi  elib  scopus
    7. Aurzada F., “Path Regularity of Gaussian Processes via Small Deviations”, Prob. Math. Stat.., 31:1 (2011), 61–78  mathscinet  zmath  isi  elib
    8. Nazarov A.I., Sheipak I.A., “Degenerate self-similar measures, spectral asymptotics and small deviations of Gaussian processes”, Bull London Math Soc, 44:1 (2012), 12–24  crossref  mathscinet  zmath  isi  elib  scopus
    9. Hu Ya., Lu F., Nualart D., “Non-Degeneracy of Some Sobolev Pseudo-Norms of Fractional Brownian Motion”, Electron. Commun. Probab., 18 (2013), 1–8  crossref  mathscinet  isi  scopus
    10. Chigansky P., Kleptsyna M., “Exact Asymptotics in Eigenproblems For Fractional Brownian Covariance Operators”, Stoch. Process. Their Appl., 128:6 (2018), 2007–2059  crossref  mathscinet  isi  scopus
    11. Lifshits M., Nazarov A., “L-2-Small Deviations For Weighted Stationary Processes”, Mathematika, 64:2 (2018), 387–405  crossref  mathscinet  zmath  isi
    12. Nazarov A.I. Nikitin Ya.Yu., “On Small Deviation Asymptotics in l-2 of Some Mixed Gaussian Processes”, 6, no. 4, 2018, 55  crossref  zmath  isi  scopus
    13. L. V. Rozovskii, “On the exact asymptotics of small deviations of $L_2$-norm for some Gaussian random fields”, Theory Probab. Appl., 63:3 (2019), 381–392  mathnet  crossref  crossref  isi  elib
    14. Ibragimov I.A. Lifshits M.A. Nazarov A.I. Zaporozhets D.N., “On the History of St. Petersburg School of Probability and Mathematical Statistics: II. Random Processes and Dependent Variables”, Vestn. St Petersb. Univ.-Math., 51:3 (2018), 213–236  crossref  isi  scopus
    15. Rozovsky L.V., “Small Ball Probabilities For Certain Gaussian Random Fields”, J. Theor. Probab., 32:2 (2019), 934–949  crossref  zmath  isi  scopus
    16. Torkhov N.A., Babak L.I., Kokolov A.A., “The Influence of Algan/Gan Heteroepitaxial Structure Fractal Geometry on Size Effects in Microwave Characteristics of Algan/Gan Hemts”, Symmetry-Basel, 11:12 (2019), 1495  crossref  isi
    17. Chigansky P., Kleptsyna M., Marushkevych D., “On the Eigenproblem For Gaussian Bridges”, Bernoulli, 26:3 (2020), 1706–1726  crossref  mathscinet  isi
    18. Chigansky P. Kleptsyna M. Marushkevych D., “Mixed Fractional Brownian Motion: a Spectral Take”, J. Math. Anal. Appl., 482:2 (2020), 123558  crossref  mathscinet  isi
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