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Small Deviations for a Family of Smooth Gaussian Processes

Author

Listed:
  • Frank Aurzada

    (Technische Universität Berlin)

  • Fuchang Gao

    (University of Idaho)

  • Thomas Kühn

    (Universität Leipzig)

  • Wenbo V. Li

    (University of Delaware)

  • Qi-Man Shao

    (Hong Kong University of Science and Technology)

Abstract

We study the small deviation probabilities of a family of very smooth self-similar Gaussian processes. The canonical process from the family has the same scaling property as standard Brownian motion and plays an important role in the study of zeros of random polynomials. Our estimates are based on the entropy method, discovered in Kuelbs and Li (J. Funct. Anal. 116:133–157, 1993) and developed further in Li and Linde (Ann. Probab. 27:1556–1578, 1999), Gao (Bull. Lond. Math. Soc. 36:460–468, 2004), and Aurzada et al. (Teor. Veroâtn. Ee Primen. 53:788–798, 2009). While there are several ways to obtain the result with respect to the L 2-norm, the main contribution of this paper concerns the result with respect to the supremum norm. In this connection, we develop a tool that allows translating upper estimates for the entropy of an operator mapping into L 2[0,1] by those of the operator mapping into C[0,1], if the image of the operator is in fact a Hölder space. The results are further applied to the entropy of function classes, generalizing results of Gao et al. (Proc. Am. Math. Soc. 138:4331–4344, 2010).

Suggested Citation

  • Frank Aurzada & Fuchang Gao & Thomas Kühn & Wenbo V. Li & Qi-Man Shao, 2013. "Small Deviations for a Family of Smooth Gaussian Processes," Journal of Theoretical Probability, Springer, vol. 26(1), pages 153-168, March.
    • Handle: RePEc:spr:jotpro:v:26:y:2013:i:1:d:10.1007_s10959-011-0380-5
    DOI: 10.1007/s10959-011-0380-5
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    References listed on IDEAS

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    1. A. A. Borovkov & P. S. Ruzankin, 2008. "On Small Deviations of Series of Weighted Random Variables," Journal of Theoretical Probability, Springer, vol. 21(3), pages 628-649, September.
    2. Nazarov, Alexander, 2009. "Log-level comparison principle for small ball probabilities," Statistics & Probability Letters, Elsevier, vol. 79(4), pages 481-486, February.
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