The arithmetic of convolutions, scale mixtures, and convolution mixtures (Q1781646)

The aim of this paper is to review and rewrite the basic properties of convolution, scale mixtures, and convolution mixtures of distributions as elementary algebraic operations. Within this formal arithmetic framework, the achievement of the paper is to derive and to retrieve known results, some of them in special conditions (e.g. when distributions are infinitely divisible measures), just as consequences of the algebraic properties on the elementary structures resulted from application of operational properties for convolutions, scale mixtures, and convolution mixtures of distributions (as measures). Clarifying examples and two interesting tables of resultant distributions as algebraic operations in various particular cases are enclosed.