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

Abstract

Let $0\le μ_1\le μ_2 \le \cdots \le μ_n$ and $δ> 0$. Samuels&#39; 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&#39;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.