Some superconcentration inequalities for extrema of stationary Gaussian Processes

TL;DR

This paper develops improved non-asymptotic tail inequalities for the extrema of stationary Gaussian processes, leveraging hypercontractive methods to better capture fluctuation rates compared to standard Gaussian concentration.

Contribution

It introduces novel superconcentration inequalities for stationary Gaussian processes using hypercontractive techniques, enhancing understanding of their fluctuation behavior.

Findings

Provides non-asymptotic tail bounds that reflect true fluctuation rates

Improves upon standard Gaussian concentration inequalities

Includes statistical examples demonstrating the inequalities' applications

Abstract

This note is concerned with concentration inequalities for extrema of stationary Gaussian processes. It provides non-asymptotic tail inequalities which fully reflect the fluctuation rate, and as such improve upon standard Gaussian concentration. The arguments rely on the hypercontractive approach developed by Chatterjee for superconcentration variance bounds. Some statistical illustrations complete the exposition.