MathDB

Samuels’ Conjecture

For 0≤μ1≤⋯≤μn0\leq\mu_1\leq\cdots\leq\mu_n and λ>∑i=1nμi\lambda>\sum_{i=1}^n\mu_i, let X1,…,XnX_1,\ldots,X_n range over independent nonnegative random variables satisfying EXi=μi\mathbb{E}X_i=\mu_i. Define Di=λ−∑k=1i−1μkD_i=\lambda-\sum_{k=1}^{i-1}\mu_k. Then inf⁡X1,…,XnP(∑i=1nXi<λ)=min⁡1≤i≤n∏j=in(1−μjDi)\inf_{X_1,\ldots,X_n}\mathbb{P}\left(\sum_{i=1}^n X_i<\lambda\right)=\min_{1\leq i\leq n}\prod_{j=i}^n\left(1-\frac{\mu_j}{D_i}\right), with the infimum attained.

Sources & referencesView supporting material

Primary source

arXiv

Additional references

inequalitiesprobabilistic-combinatoricsprobability

Progress summary

Refreshed
Claimed solved

A new preprint claims the conjecture is proved in full, but no independent confirmation has yet been found.

Samuels’ Conjecture predicts an exact product formula for an extremal probability involving independent nonnegative random variables with prescribed means. A 2017 paper records the weighted formulation and its equality cases, but explicitly assumes the conjecture rather than proving it.

Known results

  • Samuels’ conjecture is known for n=1,2,3,4n=1,2,3,4.
  • For n≥5n\geq 5, it is known in the range t≥n(n−1)t\geq n(n-1).
  • A 2026 preprint proves a special case for t≥2n+1t\geq 2n+1, improving the previous quadratic threshold.

August 2026 claimed proof

Zhi Ling’s preprint claims to prove the full conjectured product formula and show that it is attainable for arbitrary ordered means; this would settle Samuels’ Conjecture and imply Feige’s conjecture. The claim is not independently confirmed in the supplied evidence.

Current status (as of August 2026): The conjecture has a new claimed full proof in a preprint, but remains unverified; the earlier special cases are settled and the general case is not yet confirmed.

Sources

Solutions 0

No solutions have been posted yet.