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
Bound · version 2
The July bounds in two deviation ranges
Statement
For every n ≥ 1, let X₁,…,Xₙ be independent nonnegative integrable random variables with E Xᵢ ≤ 1, and put S = Σᵢ Xᵢ. For every δ > 0, when 0 < δ < 1, and 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
Connections
In this problem · publication v3 · Selected context
Feige’s probability conjecture →
Different statement · related variant · Proved
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v7
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
- 2
- SHA-256
- 35a6058bf3bb1a185e5b03d392eef25d282113de31450e8879cbbb68e8ef6482