Operation with concentration inequalities and Conjugate of parallel sum
Résumé
Following the concentration of the measure approach, we consider the transformation $Φ(Z)$ of a random variable $Z$ having a general concentration function $α$. If the transformation $Φ$ is $λ$-Lipschitz with $λ$ deterministic, the concentration function of $Φ$ is immediately deduced to be equal to $α(\cdot/λ)$. If the variations of $Φ$ are bounded by a random variable $Λ$ having a concentration function (around $0$) $β: \mathbb R_+\to \mathbb R$, this paper sets that $Φ(Z)$ has a concentration function analogous to $γ= (α^{-1} \cdot β^{-1})^{-1}$. We apply this result to generalize Hanson Wright inequality, we express $γ$ with the conjugate functions of the parallel sum of $α$ and $β$ and further consider the case where $Λ$ is the product of $n$ random variables to provide useful insights to the so-called ``multilevel concentration''.