发表机构
Department of Mathematics, Achhruram memorial College（阿赫鲁兰纪念学院数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究由对称伯努利分布与均匀分布的布尔卷积构成的单参数概率测度族，利用布尔能量变换推导显式公式与洛朗展开，计算累积量、矩及渐近估计，并分析其解析性质与缩放极限，证明该族不构成半群。
AI 中文摘要
我们引入并研究了一个由布尔加法卷积μ_t=β_t⊎U_t定义的单参数概率测度族{μ_t}_{t>0}，其中β_t是方差为t的对称伯努利分布，U_t是[-t/2,t/2]上的均匀分布。利用布尔能量变换的线性化性质，我们推导出了倒数柯西变换的显式公式，并获得了其洛朗展开。作为应用，我们计算了μ_t的布尔累积量和矩序列，建立了矩的渐近估计，并研究了包括支撑准则和布尔能量变换满足的微分方程在内的解析性质。我们进一步分析了缩放极限，证明了在适当归一化下的稳定性，并表明该族不构成布尔卷积半群。
英文摘要
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.
Comments28 pages, Comments are welcome
