On Feige's conjecture

Zipei Nie; Jiaye Wei

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arXiv · 2607.24528

On Feige's conjecture

Zipei Nie·Jiaye Wei

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Abstract

We present a short proof of Feige's conjecture: for $n$ independent nonnegative random variables with expectation one, the probability that their sum is less than $n+1$ is at least $\left(\frac{n}{n+1}\right)^n\ge \frac{1}{e}$. The proof was obtained with the assistance of GPT-5.6 Sol and builds on the recent breakthrough of Vlassis and Thomas establishing Gaffke's conjecture on the finite-sample validity of a distribution-free $p$-value. We also discuss the implications of the subsequent work of Ming, Ramdas, Shen, Wang, and Waudby-Smith.

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Zipei Nie, Jiaye Wei. 2026-07-27. On Feige's conjecture. https://arxiv.org/abs/2607.24528

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