Classes of Infinitely Divisible Distributions and Examples

Bondesson (Generalized Gamma Convolutions and Related Classes of Distributions and Densities, Lecture Notes in Statistics, vol. 76, Springer, Berlin, 1992) said "Since a lot of the standard distributions now are known to be infinitely divisible, the class of infinitely divisible distributions has perhaps partly lost its interest. Smaller classes should be more in focus." This view was presented more than two decades ago, yet has not been fully addressed. Over the last decade, many classes of infinitely divisible distributions have been studied and characterized. In this article, we summarize such "smaller classes" and try to find classes which known infinitely divisible distributions belong to, as precisely as possible.