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Teor. Veroyatnost. i Primenen., 2008, Volume 53, Issue 4, Pages 788–798 (Mi tvp2466)  

This article is cited in 12 scientific papers (total in 12 papers)

Short Communications

Small Deviations of Smooth Stationary Gaussian Processes

F. Aurzadaa, I. A. Ibragimovb, M. A. Lifshitsb, H. J. van Zandenc

a University of Sciences and Technologies
b Saint-Petersburg State University
c Vrije Universiteit Amsterdam

Abstract: We investigate the small deviation probabilities of a class of very smooth stationary Gaussian processes playing an important role in Bayesian statistical inference. Our calculations are based on the appropriate modification of the entropy method due to Kuelbs, Li, and Linde as well as on classical results about the entropy of classes of analytic functions. They also involve Tsirelson's upper bound for small deviations and shed some light on the limits of sharpness for that estimate.

Keywords: Gaussian processes, small deviations, spectral density, stationary processes.

DOI: https://doi.org/10.4213/tvp2466

Full text: PDF file (1262 kB)
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English version:
Theory of Probability and its Applications, 2009, 53:4, 697–707

Bibliographic databases:

Received: 29.03.2008

Citation: F. Aurzada, I. A. Ibragimov, M. A. Lifshits, H. J. van Zanden, “Small Deviations of Smooth Stationary Gaussian Processes”, Teor. Veroyatnost. i Primenen., 53:4 (2008), 788–798; Theory Probab. Appl., 53:4 (2009), 697–707

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. Nazarov A., “Log–level comparison principle for small ball probabilities”, Statist. Probab. Lett., 79:4 (2009), 481–486  crossref  mathscinet  zmath  isi  elib  scopus
    2. R. S. Pusev, “Asymptotics of small deviations of the Bogoliubov processes with respect to a quadratic norm”, Theoret. and Math. Phys., 165:1 (2010), 1348–1357  mathnet  crossref  crossref  adsnasa  isi
    3. Gao Fuchang, Li W.V., Wellner J.A., “How many Laplace transforms of probability measures are there?”, Proc. Amer. Math. Soc., 138:12 (2010), 4331–4344  crossref  mathscinet  zmath  isi  elib  scopus
    4. Kühn T., “Covering numbers of Gaussian reproducing kernel Hilbert spaces”, J. Complexity, 27:5 (2011), 489–499  crossref  mathscinet  zmath  isi  elib  scopus
    5. Aurzada F., “Path Regularity of Gaussian Processes via Small Deviations”, Prob. Math. Stat.., 31:1 (2011), 61–78  mathscinet  zmath  isi  elib
    6. 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
    7. A. I. Nazarov, R. S. Pusev, “Comparison theorems for the small ball probabilities of the Green Gaussian processes in weighted $L_2$-norms”, St. Petersburg Math. J., 25:3 (2014), 455–466  mathnet  crossref  mathscinet  zmath  isi  elib
    8. Kley O., “Kuelbs–Li Inequalities and Metric Entropy of Convex Hulls”, J. Theor. Probab., 26:3 (2013), 649–665  crossref  mathscinet  zmath  isi  elib  scopus
    9. Aurzada F., Gao F., Kuehn T., Li W.V., Shao Q.-M., “Small Deviations for a Family of Smooth Gaussian Processes”, J. Theor. Probab., 26:1 (2013), 153–168  crossref  mathscinet  zmath  isi  elib  scopus
    10. Karol' Andrei I., Nazarov A.I., “Small Ball Probabilities For Smooth Gaussian Fields and Tensor Products of Compact Operators”, Math. Nachr., 287:5-6 (2014), 595–609  crossref  mathscinet  zmath  isi  scopus
    11. Sottinen T., Viitasaari L., “Transfer Principle For Nth Order Fractional Brownian Motion With Applications to Prediction and Equivalence in Law”, Theory Probab. Math. Stat., 98 (2018), 188–204  isi
    12. 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
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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