A Class of Probability Distributions that is Closed with Respect to Addition as Well as Multiplication of Independent Random Variables

Abstract

Thorin’s class of generalized gamma convolutions (GGCs) is closed with respect to change in scale, weak limits, and addition of independent random variables. Here, it is shown that the GGC class also has the remarkable property of being closed with respect to multiplication of independent random variables. This novel result, which has a simple extension to symmetric distributions on (Formula presented.), has many consequences and applications. In particular, it follows that (Formula presented.) GGC implies that (Formula presented.) GGC. The latter result is used to find a large class of explicit probability functions on (Formula presented.) which are generalized negative binomial convolutions (GNBCs). The paper ends with several open problems.