AI 中文总结
研究独立非负随机变量和\(S\)小于其期望加\(\delta\)的概率下界，核心方法是结合精确狄利克雷校准定理等结果，主要贡献是给出精确小偏差不等式，肯定了Feige猜想在\(\delta\ge 1\)时的正确性。
AI 中文摘要
设\((X_1,\ldots,X_n)\)为独立非负随机变量，\(\mathbb{E} X_i\le1\)，记\(S=\sum_iX_i\)。对于\(\delta>0\)，证明了\(\mathbb{P}\left(S<\mathbb{E} S+\delta\right)\ge b_{n,\delta}\)，其中\(b_{n,\delta}\)在\(0<\delta<1\)和\(\delta\ge1\)时有不同表达式。该界对每个\(n\)和\(\delta\ge 1\)是精确的，证明结合了Vlassis和Thomas的精确狄利克雷校准定理等结果。
英文摘要
Let $(X_1,\ldots,X_n)$ be independent nonnegative random variables with $\mathbb{E} X_i\le1$, and write $S=\sum_iX_i$. For $δ>0$, we prove that \[ \mathbb{P}\left(S<\mathbb{E} S+δ\right)\ge b_{n,δ}, \] where $b_{n,δ}=δ(n/(n+δ))^n$ for $0<δ<1$ and $b_{n,δ}=(1-1/(n+δ))^n$ for $δ\ge1$. The bound is sharp for every $n$ and $δ\ge 1$. In particular, since $b_{n,δ} \ge e^{-1}$ for $δ\ge 1$, our result proves Feige's conjecture [Feige, 2004] in the affirmative for $δ\ge 1$. The proof is found by ChatGPT 5.6 Pro. It combines the exact Dirichlet calibration theorem of Vlassis and Thomas [Vlassis and Thomas, 2026], which resolves Gaffke's conjecture in statistics, with results in convex geometry including Grünbaum's centroid theorem [Grünbaum, 1960] and its generalization by Letwin and Yaskin [Letwin and Yaskin, 2024].
