Let ${\xi }_1,{\xi }_2,\dots ,{\xi }_n$ be independent random variables with $E{\xi }_i=0$ and ${\sum }_{i=1}^{n}E{\xi }_i^2=1$ and let $\Delta =\Delta ({\xi }_1,\dots ,{\xi }_n)$. Set $W={\sum }_{i=1}^n{\xi }_i$. In this note we prove that $|P(W\le \Delta )-E\Phi \left(\Delta \right)|\le 36{\sum }_{i=1}^{n}E{\xi }_i^{2}min(1,\left|{\xi }_i\right|)+24{\sum }_{i=1}^n{\Vert {\xi }_i\Vert }_2{\Vert \Delta -{\Delta }^{\left(i\right)}\Vert }_2,$ where ${\Delta }^{\left(i\right)}$ is any random variable that doesn’t depend on ${\xi }_i$. The result leads to a refined Berry–Esseen inequality for non-linear statistics $W+\Delta$ as well as a refined concentration inequality for $P(W\le {\Delta }_2)-P(W\le {\Delta }_1)$.

中文翻译：

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一个细化的随机浓度不等式

让${\xi }_1,{\xi }_2,\dots ,{\xi }_n$是独立的随机变量$乙{\xi }_{\mathrm{一世}}=0$和${\sum }_{\mathrm{一世}=1}^n乙{\xi }_{\mathrm{一世}}^2=1$然后让$\Delta =\Delta ({\xi }_1,\dots ,{\xi }_n)$. 放$W={\sum }_{\mathrm{一世}=1}^n{\xi }_{\mathrm{一世}}$. 在本笔记中，我们证明$|磷(W\le \Delta )-乙\Phi \left(\Delta \right)|\le 36{\sum }_{\mathrm{一世}=1}^n乙{\xi }_{\mathrm{一世}}^2分钟(1,\left|{\xi }_{\mathrm{一世}}\right|)+24{\sum }_{\mathrm{一世}=1}^n{\Vert {\xi }_{\mathrm{一世}}\Vert }_2{\Vert \Delta -{\Delta }^{\left(\mathrm{一世}\right)}\Vert }_2,$在哪里${\Delta }^{\left(\mathrm{一世}\right)}$是任何不依赖于的随机变量${\xi }_{\mathrm{一世}}$. 结果导致非线性统计的改进的 Berry-Esseen 不等式$W+\Delta$以及细化的集中不等式$磷(W\le {\Delta }_2)-磷(W\le {\Delta }_1)$.