Small Deviation Probability via Chaining

TL;DR

This paper extends Talagrand's lower bounds for small deviation probabilities using chaining and metric entropy, covering Gaussian and non-Gaussian stable processes with various entropy behaviors.

Contribution

It provides new bounds for small deviation probabilities for Gaussian and stable processes, exploring the limits of chaining techniques.

Findings

Extended Talagrand's lower bounds for Gaussian processes.

Derived bounds for stable processes with different entropy regimes.

Explored the limits of chaining in small deviation analysis.

Abstract

We obtain several extensions of Talagrand's lower bound for the small deviation probability using metric entropy. For Gaussian processes, our investigations are focused on processes with sub-polynomial and, respectively, exponential behaviour of covering numbers. The corresponding results are also proved for non-Gaussian symmetric stable processes, both for the cases of critically small and critically large entropy. The results extensively use the classical chaining technique; at the same time they are meant to explore the limits of this method.