Sharp Rosenthal-type inequalities for mixtures and log-concave variables
Giorgos Chasapis, Alexandros Eskenazis, Tomasz Tkocz
TL;DR
The paper addresses sharp Rosenthal-type inequalities for sums of independent random variables under structural constraints given by mixtures and log-concavity. It develops a complete picture for general mixtures, yielding exact constants in two regimes: for $2<p<4$ the extremal moment is $B^p+\|Z\|_p^pA^p$, while for $p\ge4$ the bound is governed by a Poissonized sum $\mathbb{E}|\sum_{j=1}^{\xi} \tilde{V}_j|^p$ with a precisely chosen Poisson parameter; these results recover and extend prior Gaussian-mixture work. In the log-concave setting, the extremisers are shown to lie in explicit two-point families $f_{\alpha,\gamma}$ and $g_{\alpha,\gamma}$, with $p>4$ forcing the extremisers to have the form $\mathbf{X}_j^-$ (infimum) and $\mathbf{X}_j^+$ (supremum) under moment constraints, effectively reducing to a 3-point extremal problem. The authors also extend the framework to random variables with log-concave tails, via tail-structure classes $\mathcal{G}^-$ and $\mathcal{G}^+$, and discuss directions for handling additional constraints and obtaining sharp constants in broader log-concave settings. Overall, the work provides sharp constants and extremising structures for a broad class of Rosenthal-type problems and connects to classical results by Ibragimov–Sharakhmetov and Schechtman, with implications for extremal probability under moment and shape constraints.
Abstract
We obtain Rosenthal-type inequalities with sharp constants for moments of sums of independent random variables which are mixtures of a fixed distribution. We also identify extremisers in log-concave settings when the moments of summands are individually constrained.