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 $\mathbb{E}\left\vert Y\right\vert<\infty$ and for all non-decreasing convex $\varphi:\mathbb{R}\rightarrow [0,\infty)$, $\mathbb{E}\varphi(X)\leq\mathbb{E}\varphi(Y)$, then for all $s\in\mathbb{R}$ with $\mathbb{P}\left\{Y>s\right\}\neq 0$, $\mathbb{P}\left\{X\geq\mathbb{E}\left(Y:Y>s\right)\right\}\leq\mathbb{P}\left\{Y>s\right\}$. This bound is sharp in essentially the strictest possible sense: for any such $Y$ and $s$ there exists such an $X$ with $\mathbb{P}\left\{X\geq \mathbb{E}\left(Y:Y>s\right)\right\}=\mathbb{P}\left\{Y>s\right\}$.