Optimal tail comparison under convex majorization

Abstract

Following results of Kemperman and Pinelis, we show that if $X$ and $Y$ are real valued random variables such that $\mathbb{E}\left|Y\right|<\infty$ and for all non-decreasing convex $\phi :\mathbb{R}\to [0,\infty )$, $\mathbb{E}\phi (X)\le \mathbb{E}\phi (Y)$, then for all $s\in \mathbb{R}$ with $\mathbb{P}\left\{Y>s\right\}\ne 0$, $\mathbb{P}\left\{X\ge \mathbb{E}\left(Y:Y>s\right)\right\}\le \mathbb{P}\left\{Y>s\right\}$. This bound is sharp in essentially the strictest possible sense: for any such $Y$ and $s$ there exists such an $X$ with $\mathbb{P}\left\{X\ge \mathbb{E}\left(Y:Y>s\right)\right\}=\mathbb{P}\left\{Y>s\right\}$.

9 pages. Most of the material here was originally part of arXiv:2203.12523 and/or arXiv:1812.10938 and now stands as a paper on its own