Bounds on tail probabilities for quadratic forms in dependent sub-gaussian random variables

TL;DR

This paper derives bounds on the tail probabilities of quadratic forms involving dependent sub-gaussian variables, providing tools for analyzing excess loss in linear regression with dependent data.

Contribution

It introduces new bounds on tail probabilities for quadratic forms in dependent sub-gaussian variables using Luxemburg norm estimates.

Findings

Bounds on tail probabilities for quadratic forms in dependent sub-gaussian variables.

Application to excess loss estimation in fixed design linear regression with dependent observations.

Method based on Luxemburg norm estimates of quadratic forms.

Abstract

We show bounds on tail probabilities for quadratic forms in sub-gaussian non-necessarily independent random variables. Our main tool will be estimates of the Luxemburg norms of such forms. This will allow us to formulate the above-mentioned bounds. As an example we give estimates of the excess loss in fixed design linear regression in dependent observations.