A note on the Hanson-Wright inequality for random vectors with dependencies. (English) Zbl 1328.60050

Adamczak, Radoslaw

Electron. Commun. Probab. 20, Paper No. 72, 13 p. (2015).

Summary: We prove that quadratic forms in isotropic random vectors \(X\) in \(\mathbb{R}^n\), 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.

MSC:

60E15	Inequalities; stochastic orderings
60B11	Probability theory on linear topological spaces

Keywords:

Hanson-Wright inequality; quadratic forms; concentration of measure; empirical covariance operator