On the Properties of a Bernoulli--Uniform Boolean Convolution Family

Abstract

We introduce and study a one-parameter family of probability measures $\{{\mu }_t{\}}_{t>0}$ defined by the Boolean additive convolution ${\mu }_t={\beta }_t\uplus U_t$, where ${\beta }_t$ 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 ${\mu }_t$, 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.

arXiv comment28 pages, Comments are welcome