发表机构
Technion – Israel Institute of Technology; Holon Institute of Technology (HIT)（以色列理工学院; 霍隆理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对泊松-二项分布点概率的阈值问题，发现距离2是唯一非相邻例外，推导了其阈值的显式表达式，证明结合固定均值约简与离散优化，还对实根多项式的非相邻系数反转给出严格限制。
AI 中文摘要
设S为独立伯努利随机变量的有限和，研究均值需多大时，S=k+r处的概率质量可严格超过S=k处的概率质量。对于阈值T_{k,r}:=inf{ES:P(S=k+r)>P(S=k)}，相邻情况对应经典Darroch边界k+1/(k+2)，而间隙r≥3时阈值均为k+1。距离2是唯一的非相邻例外：T_{k,2}=k+Δ_k，其中Δ_k及其阶梯优化器显式表达，且Δ_k=1/2+1/√(2k)-1/(4k)+O(k^{-3/2})。证明结合经典的固定均值约简为移位二项分布律，以及精确离散优化。该结果还对非负系数实根多项式的非相邻系数反转给出严格限制。
英文摘要
Let $S$ be a finite sum of independent Bernoulli random variables. We ask how large the mean must be before the atom at $k+r$ can strictly exceed the atom at $k$. For the threshold $T_{k,r}:=\inf\{\mathbb{E}S:\mathbb{P}(S=k+r)>\mathbb{P}(S=k)\}$, the adjacent case recovers the classical Darroch boundary $k+1/(k+2)$, while every gap $r\ge3$ has threshold $k+1$. Distance two is the unique nonadjacent exception: $T_{k,2}=k+Δ_k$, where $Δ_k$ and its staircase optimizer are explicit and $Δ_k=1/2+1/\sqrt{2k}-1/(4k)+O(k^{-3/2})$. The proof combines the classical fixed-mean reduction to shifted binomial laws with an exact discrete optimization. The result also yields sharp restrictions on nonadjacent coefficient reversals in real-rooted polynomials with nonnegative coefficients.
Comments11 pages
