Operation with Concentration Inequalities

Abstract

Following the concentration of the measure theory formalism, we consider the transformation $\Phi(Z)$ of a random variable $Z$ having a general concentration function $\alpha$. If the transformation $\Phi$ is $\lambda$-Lipschitz with $\lambda>0$ deterministic, the concentration function of $\Phi(Z)$ is immediately deduced to be equal to $\alpha(\cdot/\lambda)$. If the variations of $\Phi$ are bounded by a random variable $\Lambda$ having a concentration function (around $0$) $\beta: \mathbb R_+\to \mathbb R$, this paper sets that $\Phi(Z)$ has a concentration function analogous to the so-called parallel product of $\alpha$ and $\beta$. With this result at hand (i) we express the concentration of random vectors with independent heavy-tailed entries, (ii) given a transformation $\Phi$ with bounded $k^{\text{th}}$ differential, we express the so-called ``multilevel'' concentration of $\Phi(Z)$ as a function of $\alpha$, and the operator norms of the successive differentials up to the $k^{\text{th}}$ (iii) we obtain a heavy-tailed version of the Hanson--Wright inequality. Finally, in order to rigorously handle the algebraic operations that arise on concentration functions (parallel sums, parallel products, and non-unique pseudo-inverses), we develop at the beginning of the paper a functional framework based on maximally monotone set-valued operators, which provides a natural and coherent formalism for studying these transformations.