On the Properties of a Bernoulli--Uniform Boolean Convolution Family
Abstract
We introduce and study a one-parameter family of probability measures {μt}t>0 defined by the Boolean additive convolution μt=βtuplus Ut, where βt is the symmetric Bernoulli distribution with variance t and Ut is the uniform distribution on $[-t/2,t/2]$. Exploiting the linearization property of the Boolean energy transform, we derive an explicit formula for the reciprocal Cauchy transform and obtain its Laurent expansion. As applications, we compute the Boolean cumulants and moment sequence of μt, establish asymptotic estimates for the moments, and investigate analytic properties including support criteria and a differential equation satisfied by the Boolean energy transform. We further analyze scaling limits, prove stability under suitable normalizations, and show that the family does not form a Boolean convolution semigroup.
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BibTeX
@misc{chakraborty2026properties,
title = {{On the Properties of a Bernoulli--Uniform Boolean Convolution Family}},
author = {Sukrit Chakraborty},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06865},
eprint = {2610.06865},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06865},
}
RIS
TY - UNPB
TI - On the Properties of a Bernoulli--Uniform Boolean Convolution Family
AU - Sukrit Chakraborty
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.06865
UR - https://arxiv.org/abs/2610.06865
AB - We introduce and study a one-parameter family of probability measures {μ_t}_t>0 defined by the Boolean additive convolution μ_t=β_tuplus U_t, where β_t is the symmetric Bernoulli distribution with variance t and U_t is the uniform distribution on $[-t/2,t/2]$. Exploiting the linearization property of the Boolean energy transform, we derive an explicit formula for the reciprocal Cauchy transform and obtain its Laurent expansion. As applications, we compute the Boolean cumulants and moment sequence of μ_t, establish asymptotic estimates for the moments, and investigate analytic properties including support criteria and a differential equation satisfied by the Boolean energy transform. We further analyze scaling limits, prove stability under suitable normalizations, and show that the family does not form a Boolean convolution semigroup.
ER -