## Electronic Communications in Probability

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

#### 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

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

Rights

#### 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

#### References

• R. Adamczak, D. ChafaÃ¯, and P. Wolff, Circular law for random matrices with exchangeable entries, Random Structures & Algorithms (2015), Published online. DOI: 10.1002/rsa.20599.
• Adamczak, Radosław; Wolff, Paweł. Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order. Probab. Theory Related Fields 162 (2015), no. 3-4, 531–586.
• Adamczak, Radosław. Logarithmic Sobolev inequalities and concentration of measure for convex functions and polynomial chaoses. Bull. Pol. Acad. Sci. Math. 53 (2005), no. 2, 221–238.
• Arcones, Miguel A.; Giné, Evarist. On decoupling, series expansions, and tail behavior of chaos processes. J. Theoret. Probab. 6 (1993), no. 1, 101–122.
• Barthe, Franck; Milman, Emanuel. Transference principles for log-Sobolev and spectral-gap with applications to conservative spin systems. Comm. Math. Phys. 323 (2013), no. 2, 575–625.
• Bobkov, S. G.; GÃ¶tze, F. Discrete isoperimetric and Poincaré-type inequalities. Probab. Theory Related Fields 114 (1999), no. 2, 245–277.
• Borell, Christer. The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30 (1975), no. 2, 207–216.
• bysame, On the Taylor series of a Wiener polynomial, Seminar Notes on multiple stochastic integration, polynomial chaos and their integration. Case Western Reserve Univ., Cleveland (1984).
• Boucheron, Stéphane; Bousquet, Olivier; Lugosi, Gabor; Massart, Pascal. Moment inequalities for functions of independent random variables. Ann. Probab. 33 (2005), no. 2, 514–560.
• Chevet, S. Séries de variables aléatoires gaussiennes Ã valeurs dans $E\hat \otimes _{\varepsilon }F$. Application aux produits d'espaces de Wiener abstraits. (French) Séminaire sur la Géométrie des Espaces de Banach (1977â€“1978), Exp. No. 19, 15 pp., Ã‰cole Polytech., Palaiseau, 1978.
• Gordon, Yehoram. Some inequalities for Gaussian processes and applications. Israel J. Math. 50 (1985), no. 4, 265–289.
• Hanson, D. L.; Wright, F. T. A bound on tail probabilities for quadratic forms in independent random variables. Ann. Math. Statist. 42 1971 1079–1083.
• Hitczenko, Paweł; KwapieÅ„, Stanisław; Li, Wenbo V.; Schechtman, Gideon; Schlumprecht, Thomas; Zinn, Joel. Hypercontractivity and comparison of moments of iterated maxima and minima of independent random variables. Electron. J. Probab. 3 (1998), No. 2, 26 pp. (electronic).
• Hsu, Daniel; Kakade, Sham M.; Zhang, Tong. A tail inequality for quadratic forms of subgaussian random vectors. Electron. Commun. Probab. 17 (2012), no. 52, 6 pp.
• Koltchinskii, Vladimir. The Dantzig selector and sparsity oracle inequalities. Bernoulli 15 (2009), no. 3, 799–828. http://arxiv.org/abs/1405.2468.
• Latała, Rafał. Tail and moment estimates for some types of chaos. Studia Math. 135 (1999), no. 1, 39–53.
• Latała, Rafał. Estimates of moments and tails of Gaussian chaoses. Ann. Probab. 34 (2006), no. 6, 2315–2331.
• Latala, Rafal; Mankiewicz, Piotr; Oleszkiewicz, Krzysztof; Tomczak-Jaegermann, Nicole. Banach-Mazur distances and projections on random subgaussian polytopes. Discrete Comput. Geom. 38 (2007), no. 1, 29–50.
• Ledoux, Michel. On Talagrand's deviation inequalities for product measures. ESAIM Probab. Statist. 1 (1995/97), 63–87 (electronic).
• Ledoux, Michel. The concentration of measure phenomenon. Mathematical Surveys and Monographs, 89. American Mathematical Society, Providence, RI, 2001. x+181 pp. ISBN: 0-8218-2864-9
• Ledoux, Michel; Talagrand, Michel. Probability in Banach spaces. Isoperimetry and processes. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 23. Springer-Verlag, Berlin, 1991. xii+480 pp. ISBN: 3-540-52013-9
• Maggi, Francesco. Sets of finite perimeter and geometric variational problems. An introduction to geometric measure theory. Cambridge Studies in Advanced Mathematics, 135. Cambridge University Press, Cambridge, 2012. xx+454 pp. ISBN: 978-1-107-02103-7
• Meckes, Mark W.; Szarek, Stanisław J. Concentration for noncommutative polynomials in random matrices. Proc. Amer. Math. Soc. 140 (2012), no. 5, 1803–1813.
• Paulin, Daniel. The convex distance inequality for dependent random variables, with applications to the stochastic travelling salesman and other problems. Electron. J. Probab. 19 (2014), no. 68, 34 pp.
• Rauhut, Holger; Romberg, Justin; Tropp, Joel A. Restricted isometries for partial random circulant matrices. Appl. Comput. Harmon. Anal. 32 (2012), no. 2, 242–254.
• Rudelson, Mark; Vershynin, Roman. Hanson-Wright inequality and sub-Gaussian concentration. Electron. Commun. Probab. 18 (2013), no. 82, 9 pp.
• Samson, Paul-Marie. Concentration of measure inequalities for Markov chains and $\Phi$-mixing processes. Ann. Probab. 28 (2000), no. 1, 416–461.
• Sudakov, V. N.; CirelÊ¹son, B. S. Extremal properties of half-spaces for spherically invariant measures. (Russian) Problems in the theory of probability distributions, II. Zap. NauÄn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 41 (1974), 14–24, 165.
• Talagrand, Michel. An isoperimetric theorem on the cube and the Kintchine-Kahane inequalities. Proc. Amer. Math. Soc. 104 (1988), no. 3, 905–909.
• Talagrand, Michel. Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Ã‰tudes Sci. Publ. Math. No. 81 (1995), 73–205.
• Talagrand, Michel. New concentration inequalities in product spaces. Invent. Math. 126 (1996), no. 3, 505–563.
• Van Vu and Ke Wang, Random weighted projections, random quadratic forms and random eigenvectors, Random Structures & Algorithms (2014), Published online. DOI: 10.1002/rsa.20561.
• Wright, F. T. A bound on tail probabilities for quadratic forms in independent random variables whose distributions are not necessarily symmetric. Ann. Probability 1 (1973), no. 6, 1068–1070.