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VOL. 9 | 2013 Around Nemirovski’s inequality
Pascal Massart, Raphaël Rossignol

Editor(s) M. Banerjee, F. Bunea, J. Huang, V. Koltchinskii, M. H. Maathuis

Abstract

Nemirovski’s inequality states that given independent and centered at expectation random vectors $X_{1},\ldots,X_{n}$ with values in $\ell^p(\mathbb{R}^d)$, there exists some constant $C(p,d)$ such that

\[\mathbb{E}\Vert S_n\Vert _p^2\le C(p,d)\sum_{i=1}^{n}\mathbb{E}\Vert X_i\Vert _p^2.\]

Furthermore $C(p,d)$ can be taken as $\kappa(p\wedge \log(d))$. Two cases were studied further in [ Am. Math. Mon. 117(2) (2010) 138–160]: general finite-dimensional Banach spaces and the special case $\ell^{\infty}(\mathbb{R}^{d})$. We show that in these two cases, it is possible to replace the quantity $\sum_{i=1}^n\mathbb{E}\Vert X_i\Vert _p^2$ by a smaller one without changing the order of magnitude of the constant when $d$ becomes large. In the spirit of [ Am. Math. Mon. 117(2) (2010) 138–160], our approach is probabilistic. The derivation of our version of Nemirovski’s inequality indeed relies on concentration inequalities.

Information

Published: 1 January 2013
First available in Project Euclid: 8 March 2013

zbMATH: 1355.60010
MathSciNet: MR3202638

Digital Object Identifier: 10.1214/12-IMSCOLL918

Keywords: Concentration inequalities , Efron-Stein’s inequality , high dimensional Banach space , Maximal inequalities , Nemirovski’s inequality

Rights: Copyright © 2010, Institute of Mathematical Statistics

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