AI 中文总结
本文通过“cap move”操作给出林氏关于塞缪尔猜想证明的另一种表述，该证明直接作用于有限支撑分布，避免归约至伯努利情形。
AI 中文摘要
设0≤μ₁≤μ₂≤…≤μₙ，δ>0。塞缪尔猜想指出，若X₁,…,Xₙ为独立非负随机变量，满足E[Xᵢ]=μᵢ，则P(∑ᵢ₌₁ⁿXᵢ<δ+∑ᵢ₌₁ⁿμᵢ)≥min₁≤i≤n∏ⱼ₌ᵢⁿ(1−μⱼ/(δ+∑ₖ₌ᵢⁿμₖ))。该猜想近期已由林证明。本文通过名为“cap move”（上移）的操作，给出林氏证明的另一种表述，此证明直接作用于有限支撑分布，避免了向伯努利情形的归约。
英文摘要
Let $0\le μ_1\le μ_2 \le \cdots \le μ_n$ and $δ> 0$. Samuels' conjecture claims that if $X_1,\dots,X_n$ are independent non-negative random variables with $\mathbb{E}[X_i] = μ_i$, then $$ \mathbb{P}\left( \sum_{i=1}^n X_i < δ+ \sum_{i=1}^n μ_i \right) \ge \min_{1\le i\le n} \prod_{j=i}^n \left(1-\frac{μ_j}{δ+ \sum_{k=i}^n μ_k}\right).$$ This conjecture was recently proved by Ling. In this note, we provide an alternative presentation of Ling's proof via an operation called the \emph{cap move}. This proof works directly with finitely supported distributions and avoids the reduction to the Bernoulli case.
