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A Sharp Refinement of Samuels' Inequality via Cap Moves
Abstract
Let 0≤ μ1≤ μ2 ≤ ⋯ ≤ μn and δ> 0. Samuels' conjecture states that if X1,…,Xn are independent non-negative random variables with 𝔼[Xi] = μi, then ℙ( ∑i=1n Xi < δ+ ∑i=1n μi ) ≥ min1≤ i≤ n ∏j=in (1−μj/(δ+ ∑k=in μk)). This conjecture was recently proved by Ling. In this note, we prove a refinement of Samuels' inequality via an operation called the \emph{cap move}. Our refined inequality is sharp, with equality attained for a strictly larger class of distributions than in the original inequality.
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BibTeX
@misc{han2026sharp,
title = {{A Sharp Refinement of Samuels' Inequality via Cap Moves}},
author = {Yanjun Han and Zipei Nie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.22816},
eprint = {2608.22816},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22816},
}
RIS
TY - UNPB
TI - A Sharp Refinement of Samuels' Inequality via Cap Moves
AU - Yanjun Han
AU - Zipei Nie
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2608.22816
UR - https://arxiv.org/abs/2608.22816
AB - Let 0≤ μ_1≤ μ_2 ≤ ⋯ ≤ μ_n and δ> 0. Samuels' conjecture states that if X_1,…,X_n are independent non-negative random variables with 𝔼[X_i] = μ_i, then ℙ( ∑_i=1^n X_i < δ+ ∑_i=1^n μ_i ) ≥ min_1≤ i≤ n ∏_j=i^n (1−μ_j/(δ+ ∑_k=i^n μ_k)). This conjecture was recently proved by Ling. In this note, we prove a refinement of Samuels' inequality via an operation called the \emph{cap move}. Our refined inequality is sharp, with equality attained for a strictly larger class of distributions than in the original inequality.
ER -