Small Deviations for a Family of Smooth Gaussian Processes
Li, Wenbo V.
|Source||Journal of theoretical probability , v. 26, (1), March 2013, p. 153-168|
|Summary||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. VeroA cent tn. 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 Holder 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).|
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