Small deviations for a family of smooth Gaussian processes. (English) Zbl 1297.60022
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 by J. Kuelbs and W. V. Li [J. Funct. Anal. 116, No. 1, 133–157 (1993; Zbl 0799.46053)] and developed further by W. V. Li and W. Linde [Ann. Probab. 27, No. 3, 1556–1578 (1999; Zbl 0983.60026)], by F.-C. Gao [Bull. Lond. Math. Soc. 36, No. 4, 460–468 (2004; Zbl 1060.41023)], and by F. Aurzada et al. [Theory Probab. Appl. 53, No. 4, 697–707 (2009); translation from Teor. Veroyatn. Primen. 53, No. 4, 788–797 (2008; Zbl 1192.60055)]. 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 F.-C. Gao et al. [Proc. Am. Math. Soc. 138, No. 12, 4331–4344 (2010; Zbl 1206.60020)].
Our estimates are based on the entropy method, discovered by J. Kuelbs and W. V. Li [J. Funct. Anal. 116, No. 1, 133–157 (1993; Zbl 0799.46053)] and developed further by W. V. Li and W. Linde [Ann. Probab. 27, No. 3, 1556–1578 (1999; Zbl 0983.60026)], by F.-C. Gao [Bull. Lond. Math. Soc. 36, No. 4, 460–468 (2004; Zbl 1060.41023)], and by F. Aurzada et al. [Theory Probab. Appl. 53, No. 4, 697–707 (2009); translation from Teor. Veroyatn. Primen. 53, No. 4, 788–797 (2008; Zbl 1192.60055)]. 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 F.-C. Gao et al. [Proc. Am. Math. Soc. 138, No. 12, 4331–4344 (2010; Zbl 1206.60020)].
Keywords:
small ball probability; small deviation probability; Gaussian process; self-similar process; metric entropyReferences:
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