Year: 2026
Type: preprint
Abstract: We introduce and study a one-parameter family of probability measures $\{μ_t\}_{t>0}$ defined by the Boolean additive convolution $μ_t=β_t\uplus U_t$, 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.
Source: arXiv (Cornell University)
Author: Chakraborty, Sukrit
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221020796