Two moments suffice for Bessel approximation: conditioned differences, contingency tables, and probabilistic divide-and-conquer
Abstract
We develop Stein's method, with explicit constants throughout, for the discrete Bessel distribution: the law of a Poisson variable given its difference from an independent Poisson variable. The Bessel family is characterized by a birth-death operator with quadratic death rate n(n+k) and as the exact fixed-point set of a centered quadratic-bias transform, and we prove a dependency expansion of Arratia-Goldstein-Gordon type: two joint moments of a column array control the total-variation distance from the conditioned row sum to a matched Bessel law. Applied to the row-constrained Fisher-Yates ensemble on 2× c contingency tables -- independent Bernoulli or capped-Poisson entries conditioned on the row-sum difference, column sums free -- this yields a limit theorem -- which upgrades to the entire conditioned table at the same two-moment error -- a completely explicit enumeration estimate, and an exact divide-and-conquer sampler with certified acceptance. The entire apparatus -- operator, expansion, enumeration, sampler -- generalizes to r× c tables, where conditioning on all successive row differences produces a hyper-Bessel law. Verification scripts are included as ancillary files.
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BibTeX
@misc{desalvo2026moments,
title = {{Two moments suffice for Bessel approximation: conditioned differences, contingency tables, and probabilistic divide-and-conquer}},
author = {Stephen DeSalvo},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00263},
eprint = {2611.00263},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00263},
}
RIS
TY - UNPB
TI - Two moments suffice for Bessel approximation: conditioned differences, contingency tables, and probabilistic divide-and-conquer
AU - Stephen DeSalvo
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00263
UR - https://arxiv.org/abs/2611.00263
AB - We develop Stein's method, with explicit constants throughout, for the discrete Bessel distribution: the law of a Poisson variable given its difference from an independent Poisson variable. The Bessel family is characterized by a birth-death operator with quadratic death rate n(n+k) and as the exact fixed-point set of a centered quadratic-bias transform, and we prove a dependency expansion of Arratia-Goldstein-Gordon type: two joint moments of a column array control the total-variation distance from the conditioned row sum to a matched Bessel law. Applied to the row-constrained Fisher-Yates ensemble on 2× c contingency tables -- independent Bernoulli or capped-Poisson entries conditioned on the row-sum difference, column sums free -- this yields a limit theorem -- which upgrades to the entire conditioned table at the same two-moment error -- a completely explicit enumeration estimate, and an exact divide-and-conquer sampler with certified acceptance. The entire apparatus -- operator, expansion, enumeration, sampler -- generalizes to r× c tables, where conditioning on all successive row differences produces a hyper-Bessel law. Verification scripts are included as ancillary files.
ER -