A note on the Hanson-Wright inequality for random vectors with dependencies

We prove that quadratic forms in isotropic random vectors X in Rⁿ, possessing the convex concentration property with constant K, satisfy the Hanson-Wright inequality with constant $CK$, where C is an absolute constant, thus eliminating the logarithmic (in the dimension) factors in a recent estimate by Vu and Wang. We also show that the concentration inequality for all Lipschitz functions implies a uniform version of the Hanson-Wright inequality for suprema of quadratic forms (in the spirit of the inequalities by Borell, Arcones-Giné and Ledoux-Talagrand). Previous results of this type relied on stronger isoperimetric properties of X and in some cases provided an upper bound on the deviations rather than a concentration inequality.In the last part of the paper we show that the uniform version of the Hanson-Wright inequality for Gaussian vectors can be used to recover a recent concentration inequality for empirical estimators of the covariance operator of B-valued Gaussian variables due to Koltchinskii and Lounici.