Small deviations for a family of smooth Gaussian processes

TL;DR

This paper investigates small deviation probabilities of smooth Gaussian processes, especially in the supremum norm, introducing new tools to relate entropy estimates between different function spaces and applying these to function class entropy.

Contribution

It develops a novel method to connect entropy estimates in $L_2$ and $C[0,1]$ spaces for smooth Gaussian processes, enhancing understanding of their small deviations.

Findings

Established entropy bounds for smooth Gaussian processes in supremum norm.

Extended entropy estimates to broader function classes.

Provided tools for translating entropy estimates between function spaces.

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 (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 $L_2$ 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 $L_2[0,1]$ by those of the operator mapping into $C[0,1]$, if the image of the operator is in fact a H\"older space. The results are further applied to the entropy of function classes, generalizing results of Gao et al. (2010).