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The Berry--Esseen estimate in the Boolean central limit theorem
Abstract
We estimate the Lévy distance to the symmetric Bernoulli distribution for normalized Boolean convolution powers of a probability measure with mean zero and variance one. If the absolute moment of order r is finite, we obtain the upper bounds o(n−(r−2)/6) for 2<r<5 and O(n−1/2) for r≥5. The proof relies on an inequality that controls the Lévy distance to the Bernoulli law by the values of the reciprocal Cauchy transform at two points. For each r≥2, we also construct a single fixed distribution whose absolute moments are finite exactly up to order r, and show that the exponent min{(r−2)/6,1/2} cannot be improved. Under only a finite second moment assumption, the convergence can be arbitrarily slow.
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BibTeX
@misc{makoto2026berryesseen,
title = {{The Berry--Esseen estimate in the Boolean central limit theorem}},
author = {Maejima Makoto and Noriyoshi Sakuma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39743},
eprint = {2610.39743},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39743},
}
RIS
TY - UNPB
TI - The Berry--Esseen estimate in the Boolean central limit theorem
AU - Maejima Makoto
AU - Noriyoshi Sakuma
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39743
UR - https://arxiv.org/abs/2610.39743
AB - We estimate the Lévy distance to the symmetric Bernoulli distribution for normalized Boolean convolution powers of a probability measure with mean zero and variance one. If the absolute moment of order r is finite, we obtain the upper bounds o(n^−(r−2)/6) for 2<r<5 and O(n^−1/2) for r≥5. The proof relies on an inequality that controls the Lévy distance to the Bernoulli law by the values of the reciprocal Cauchy transform at two points. For each r≥2, we also construct a single fixed distribution whose absolute moments are finite exactly up to order r, and show that the exponent min{(r−2)/6,1/2} cannot be improved. Under only a finite second moment assumption, the convergence can be arbitrarily slow.
ER -