2010/09/28 by Frank Aurzada, Aurzada, Frank, Fuchang Gao +7
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Statistical Methods and Inference #Stochastic processes and financial applications #math.PR
paper · pdf · doi:10.48550/arxiv.1009.5580
16p, to appear in: Journal of Theoretical Probability
openalex publication_date 2010/09/28 · arxiv created 2011/08/17 · arxiv updated 2011/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (1992) and developed further in Li and Linde (1999), Gao (2004), and Aurzada et al. (2009). While there are several ways to obtain the result w.r.t. the L2 norm, the main contribution of this paper concerns the result w.r.t. the supremum norm. In this connection, we develop a tool that allows to translate upper estimates for the entropy of an operator mapping into L2[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. (2010).