On Small deviations of Gaussian processes using majorizing measures

TL;DR

This paper explores small deviations in Gaussian processes, demonstrating that processes with similar entropy can have vastly different behaviors, and introduces a sharp lower bound using majorizing measures.

Contribution

It provides new examples of Gaussian processes with identical entropy but different small deviations and establishes a general, sharp lower bound via majorizing measures.

Findings

Examples of Gaussian processes with same entropy but different small deviations

A sharp lower bound for small deviations using majorizing measures

Application of bounds to Gaussian sequences and ultrametric processes

Abstract

We give two examples of periodic Gaussian processes, having entropy numbers of exactly same order but radically different small deviations. Our construction is based on classical Knopp's result yielding of existence of continuous nowhere differentiable functions, and more precisely on Loud's functions. We also obtain a general lower bound for small deviations using the majorizing measure method. We show on examples that our bound is sharp. We also apply it to Gaussian independent sequences and to the generic class of ultrametric Gaussian processes. 60F15, 60G50 ; Secondary: 60F05