Open Access
November 2018 Concentration inequalities for separately convex functions
Antoine Marchina
Bernoulli 24(4A): 2906-2933 (November 2018). DOI: 10.3150/17-BEJ949


We provide new comparison inequalities for separately convex functions of independent random variables. Our method is based on the decomposition in Doob martingale. However, we only impose that the martingale increments are stochastically bounded. For this purpose, building on the results of Bentkus (Lith. Math. J. 48 (2008) 237–255; Lith. Math. J. 48 (2008) 137–157; Bounds for the stop loss premium for unbounded risks under the variance constraints (2010) Preprint), we establish comparison inequalities for random variables stochastically dominated from below and from above. We illustrate our main results by showing how they can be used to derive deviation or moment inequalities for functions which are both separately convex and separately Lipschitz, for weighted empirical distribution functions, for suprema of randomized empirical processes and for chaos of order two.


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Antoine Marchina. "Concentration inequalities for separately convex functions." Bernoulli 24 (4A) 2906 - 2933, November 2018.


Received: 1 September 2016; Revised: 1 April 2017; Published: November 2018
First available in Project Euclid: 26 March 2018

zbMATH: 06853269
MathSciNet: MR3779706
Digital Object Identifier: 10.3150/17-BEJ949

Keywords: Concentration inequalities , Deviation inequalities , Generalized moments , martingale method , suprema of empirical processes

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

Vol.24 • No. 4A • November 2018
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