Small Deviations for a Family of Smooth Gaussian Processes

Frank Aurzada (), Fuchang Gao (), Thomas Kühn (), Wenbo V. Li () and Qi-Man Shao ()
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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

Journal of Theoretical Probability, 2013, vol. 26, issue 1, 153-168

Abstract: 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).

Keywords: Small ball probability; Small deviation probability; Gaussian process; Self-similar process; Metric entropy; 60G15; 60F99; 60F10 (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s10959-011-0380-5

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