A Cap-Move Reformulation of Ling's Proof of Samuels' Conjecture

Let $0\le {\mu }_1\le {\mu }_2\le \cdots \le {\mu }_n$ and $\delta >0$. Samuels’ conjecture claims that if $X_1,\dots ,X_n$ are independent non-negative random variables with $\mathbb{E}[X_i]={\mu }_i$, then $$\mathbb{P}(\sum _{i=1}^n{X}_i<\delta +\sum _{i=1}^n{\mu }_i)\ge \underset{1\le i\le n}{\min }\prod _{j=i}^n(1-\frac{{\mu }_j}{\delta +\sum _{k=i}^n{\mu }_k})\mathrm{.}$$ 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.