Uniform Hanson-Wright Type Deviation Inequalities for $\alpha$-Subexponential Random Vectors

This paper is devoted to uniform versions of the Hanson-Wright inequality for a random vector with independent centered $\alpha$-subexponential entries, $0<\alpha \le 1$. Our method relies upon a novel decoupling inequality and a comparison of weak and strong moments. As an application, we use the derived inequality to prove the restricted isometry property of partial random circulant matrices generated by standard $\alpha$-subexponential random vectors, $0<\alpha \le 1$.