The July bounds in two deviation ranges

Statement

Under the same hypotheses, for every δ > 0, $\mathbb P(S<\mathbb ES+\delta)\ge\delta\left(\frac n{n+\delta}\right)^n$ when 0 < δ < 1, and $\mathbb P(S<\mathbb ES+\delta)\ge\left(1-\frac1{n+\delta}\right)^n$ when δ ≥ 1. The latter bound is sharp for each fixed n and δ ≥ 1.

Assumptions

Independent nonnegative integrable variables, each mean at most one.
Positive n and δ; the event uses a strict inequality.

Scope

The first branch is a lower bound, without a sharpness claim. Nie–Wei prove the first expression for unit means and 0 < δ ≤ 1; shifting each variable by 1 minus its mean extends that statement to means at most one.

Evidence

Source reports

Proved

Two parameter-range bounds proved in the cited paper ↗

Lower bound δ(n/(n+δ))^n for 0 < δ < 1, and (1−1/(n+δ))^n for δ ≥ 1. Sharpness is asserted only in the second range.

Sharp small-deviation inequalities for sums of independent nonnegative random variables · Theorem 1.1, Corollary 1.2 and AI disclosure →

Source dated Jul 27, 2026 · Observed Oct 13, 2026

Connections

In this problem · publication v2

Feige’s probability conjecture →

Different statement · related variant · Proved

Relationship not independently reviewed. This concerns its use in this problem.

Sources

On Feige’s conjecture →

Theorem 1.3 and AI acknowledgement; arXiv submitted 27 July, HTML manuscript dated 24 August → · Original passage ↗
Recorded wording
Gives a separate argument for the unit-slack inequality and the bound δ(n/(n+δ))^n for 0 < δ ≤ 1 under unit means.
paraphrase · OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.

Zipei Nie · Jiaye Wei · Preprint · Publication date Jul 27, 2026

Revision & attribution

Source revision: 2026-07-27. Observed Oct 13, 2026.

Nie and Wei credit GPT-5.6 Sol for exploration and discovery, state that they checked the arguments and authored the text, and retain responsibility. Their argument uses the Vlassis–Thomas calibration theorem.

Sharp small-deviation inequalities for sums of independent nonnegative random variables →

Theorem 1.1, Corollary 1.2 and AI disclosure → · Original passage ↗
Recorded wording
Proves two parameter-range lower bounds, with the finite-n bound sharp when δ ≥ 1.
paraphrase · OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.

Weibo Fu · Yanjun Han · Guanyang Wang · Jun Yan · Peng Zhang · Zhengqing Zhou · Preprint · Publication date Jul 27, 2026

Revision & attribution

Source revision: 2026-07-27. Observed Oct 13, 2026.

The authors report GPT-5.6 Pro assistance in proof discovery; the calibration theorem of Vlassis and Thomas and Grünbaum-type inequalities are mathematical inputs, not authors of this paper.

History

Statement versions

Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.

Assessment history · 1

Assessments retained in these publication records. Earlier statement versions retain their own source histories.

Proved

Two parameter-range bounds proved in the cited paper ↗

Lower bound δ(n/(n+δ))^n for 0 < δ < 1, and (1−1/(n+δ))^n for δ ≥ 1. Sharpness is asserted only in the second range.

Source dated Jul 27, 2026 · Observed Oct 13, 2026 · Source passage →

Record details

Claim ID: claim-feige-july-deviation-bounds
Statement version: 1
SHA-256: 0b539068ec6c7bfc2b100d555fc20523e2a4d6d55fd2cf9256496cd80084bd6f

Publication records

Feige’s probability conjecture · publication v2 · JSON →