Certain families of probability distribution functions maintain their infinite divisibility under repeated mixing and convolution. Examples on the continuum and lattice are given. The main tools used are Polya's criteria and the properties of log-convexity and complete monotonicity. Some light is shed on the relationship between these two properties.
J. Keilson. F. W. Steutel. "Families of Infinitely Divisible Distributions Closed Under Mixing and Convolution." Ann. Math. Statist. 43 (1) 242 - 250, February, 1972. https://doi.org/10.1214/aoms/1177692717