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2015 A note on the Hanson-Wright inequality for random vectors with dependencies
Radoslaw Adamczak
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Electron. Commun. Probab. 20: 1-13 (2015). DOI: 10.1214/ECP.v20-3829


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.


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Radoslaw Adamczak. "A note on the Hanson-Wright inequality for random vectors with dependencies." Electron. Commun. Probab. 20 1 - 13, 2015.


Accepted: 8 October 2015; Published: 2015
First available in Project Euclid: 7 June 2016

zbMATH: 1328.60050
MathSciNet: MR3407216
Digital Object Identifier: 10.1214/ECP.v20-3829

Primary: 60E15
Secondary: 60B11

Keywords: concentration of measure , empirical covariance operator , Hanson-Wright inequality , Quadratic forms


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