## Bernoulli

• Bernoulli
• Volume 24, Number 4A (2018), 2906-2933.

### Concentration inequalities for separately convex functions

Antoine Marchina

#### Abstract

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.

#### Article information

Source
Bernoulli, Volume 24, Number 4A (2018), 2906-2933.

Dates
Received: September 2016
Revised: April 2017
First available in Project Euclid: 26 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.bj/1522051229

Digital Object Identifier
doi:10.3150/17-BEJ949

Mathematical Reviews number (MathSciNet)
MR3779706

Zentralblatt MATH identifier
06853269

#### Citation

Marchina, Antoine. Concentration inequalities for separately convex functions. Bernoulli 24 (2018), no. 4A, 2906--2933. doi:10.3150/17-BEJ949. https://projecteuclid.org/euclid.bj/1522051229

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