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

Abstract

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 $$ \mathbb{P}\left( \sum_{i=1}^n X_i < \delta + \sum_{i=1}^n \mu_i \right) \ge \min_{1\le i\le n} \prod_{j=i}^n \left(1-\frac{\mu_j}{\delta + \sum_{k=i}^n \mu_k}\right).$$ 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.