Concentration inequalities for polynomials in α-sub-exponential random variables

We derive multi-level concentration inequalities for polynomials in independent random variables with an alpha-sub-exponential tail decay. A particularly interesting case is given by quadratic forms f (X-1, ..., X-n) = < X, AX >, 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 them are generalizations of some results proven by Rudelson and 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 and concentration inequalities for the distance between a random vector and a fixed subspace.