Sharp deviation bounds for quadratic forms

TL;DR

This paper derives precise, non-asymptotic deviation bounds for quadratic forms of random vectors with finite exponential moments, applicable under general conditions and similar to Gaussian cases.

Contribution

It provides sharp, explicit deviation inequalities for quadratic forms without assuming specific structure, extending the scope of existing bounds.

Findings

Bounds are exact and non-asymptotic.

Constants in bounds are explicit.

Results are similar to Gaussian quadratic forms.

Abstract

This note presents sharp inequalities for deviation probability of a general quadratic form of a random vector \(\xiv\) with finite exponential moments. The obtained deviation bounds are similar to the case of a Gaussian random vector. The results are stated under general conditions and do not suppose any special structure of the vector \(\xiv\). The obtained bounds are exact (non-asymptotic), all constants are explicit and the leading terms in the bounds are sharp.