Why Distance Two Is Exceptional for Poisson-Binomial Point Probabilities

Abstract

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{𝔼S:ℙ(S=k+r)>ℙ(S=k)}, the adjacent case recovers the classical Darroch boundary k+1/(k+2), while every gap r≥3 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/√(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.

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BibTeX RIS

BibTeX

@misc{kleiner2026distance,
  title = {{Why Distance Two Is Exceptional for Poisson-Binomial Point Probabilities}},
  author = {Igor Kleiner},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10664},
  eprint = {2610.10664},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10664},
}

RIS

TY  - UNPB
TI  - Why Distance Two Is Exceptional for Poisson-Binomial Point Probabilities
AU  - Igor Kleiner
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10664
UR  - https://arxiv.org/abs/2610.10664
AB  - 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{𝔼S:ℙ(S=k+r)>ℙ(S=k)}, the adjacent case recovers the classical Darroch boundary k+1/(k+2), while every gap r≥3 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/√(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.
ER  -

Abstract

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