Optimal tail comparison under convex majorization

TL;DR

This paper establishes a sharp bound on the tail probability of a random variable X under convex majorization conditions relative to Y, extending previous results and characterizing the optimal tail comparison.

Contribution

It generalizes tail comparison bounds under convex majorization, providing the sharpest possible inequality and explicit constructions achieving equality.

Findings

The bound is sharp and essentially the strictest possible.

For any Y and s, there exists an X attaining the bound equality.

The result extends Kemperman and Pinelis's earlier work.

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 $\mathbb{P}\left\{X\geq \mathbb{E}\left(Y:Y>s\right)\right\}=\mathbb{P}\left\{Y>s\right\}$.