# Concentration inequalities for polynomials in $\alpha$-sub-exponential random variables

## TL;DR

This paper develops new multi-level concentration inequalities for polynomial functions of independent $ ext{alpha}$-sub-exponential random variables, including quadratic forms, with applications to regression, random projections, and probabilistic bounds.

## Contribution

It introduces novel concentration inequalities for polynomials in $ ext{alpha}$-sub-exponential variables, extending known results from sub-Gaussian to broader tail decay classes.

## Key findings

- Derived Hanson-Wright-type inequalities with explicit matrix norm dependence.
- Established two-level concentration inequalities for quadratic forms.
- Extended Rudelson-Vershynin results to $ ext{alpha}$-sub-exponential variables.

## Abstract

In this work we derive multi-level concentration inequalities for polynomial functions in independent random variables with a $\alpha$-sub-exponential tail decay. A particularly interesting case is given by quadratic forms $f(X_1, \ldots, X_n) = \langle X,A X \rangle$, for which we prove Hanson-Wright-type inequalities with explicit dependence on various norms of the matrix $A$. A consequence of these inequalities is a two-level concentration inequality for quadratic forms in $\alpha$-sub-exponential random variables, such as quadratic Poisson chaos. We provide various applications of these inequalities. Among these are generalizations the results given by Rudelson-Vershynin from sub-Gaussian to $\alpha$-sub-exponential random variables, i. e. concentration of the Euclidean norm of the linear image of a random vector, small ball probability estimates and concentration inequalities for the distance between a random vector and a fixed subspace. Moreover, we obtain concentration inequalities for the excess loss in a fixed design linear regression and the norm of a randomly projected random vector.