Electronic Communications in Probability

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

Radoslaw Adamczak

Full-text: Open access

Abstract

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.

Article information

Source
Electron. Commun. Probab. Volume 20 (2015), paper no. 72, 13 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465320999

Digital Object Identifier
doi:10.1214/ECP.v20-3829

Mathematical Reviews number (MathSciNet)
MR3407216

Zentralblatt MATH identifier
1328.60050

Subjects
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60B11: Probability theory on linear topological spaces [See also 28C20]

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

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Adamczak, Radoslaw. A note on the Hanson-Wright inequality for random vectors with dependencies. Electron. Commun. Probab. 20 (2015), paper no. 72, 13 pp. doi:10.1214/ECP.v20-3829. https://projecteuclid.org/euclid.ecp/1465320999


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