论文标题
在凸大大化下的最佳尾巴比较
Optimal tail comparison under convex majorization
论文作者
论文摘要
以下Kemperman和Pinelis的结果，我们表明，如果$ x $和$ y $是真正有价值的随机变量，以至于$ \ mathbb {e} \ left \ welet \ vert y \ right \ right \ right \ vert <\ infty $，并且对于所有非decreasing convex $φ $ \ mathbb { $ \ mathbb {p} \ left \ {x \ geq \ mathbb {e} \ left（y：y：y> s \ right）\ right \} \ leq \ leq \ leq \ mathbb {p} \ left \ left \ left \ left \ {y> s \ s \ \ \ right \ right \} $。从本质上讲，这种界限是最严格的：对于任何这样的$ y $和$ s $，都存在$ x $，$ \ mathbb {p} \ left \ {x \ geq \ geq \ geq \ geq \ geq \ geq \ mathbb {e} \ left（y：y：y：y> s \ s \ right \ right \ right \ right \ right \ right \ right \ right \ mathbb}
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 $φ:\mathbb{R}\rightarrow [0,\infty)$, $\mathbb{E}φ(X)\leq\mathbb{E}φ(Y)$, 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 $\mathbb{P}\left\{X\geq \mathbb{E}\left(Y:Y>s\right)\right\}=\mathbb{P}\left\{Y>s\right\}$.
