Duals of convolution thinned relationships

In a recent article, J. Peyhardi gives a number of novel results related to quasi Pólya thinning which encompass a number of important mixture relationships between univariate discrete distributions. In this note, I explore the duals of the general results on convolution thinning given in Peyhardi's Theorem 1 in order to obtain new relationships and to gain new insights into old relationships. Some consequences—for integer‐valued autoregressive processes—and analogues—in the continuous case—are noted.