Small Deviations of Smooth Stationary Gaussian Processes

TL;DR

This paper analyzes the small deviation probabilities of smooth stationary Gaussian processes, using entropy methods and classical analytic function results, with implications for Bayesian inference and bounds on deviations.

Contribution

It introduces a modified entropy method tailored for smooth Gaussian processes and explores the sharpness of Tsirelson's upper bound in this context.

Findings

Derived new bounds for small deviation probabilities

Extended entropy methods to analytic function classes

Provided insights into the limits of Tsirelson's upper bound

Abstract

We investigate the small deviation probabilities of a class of very smooth stationary Gaussian processes playing an important role in Bayesian statistical inference. Our calculations are based on the appropriate modification of the entropy method due to Kuelbs, Li, and Linde as well as on classical results about the entropy of classes of analytic functions. They also involve Tsirelson's upper bound for small deviations and shed some light on the limits of sharpness for that estimate.