Maceda (1948) characterized the mixed Poisson distributions that are Poisson-stopped-sum distributions based on the mixing distribution. In an alternative characterization of the same set of distributions here the Poisson-stopped-sum distributions that are mixed Poisson distributions is proved to be the set of Poisson-stopped-sums of either a mixture of zero-truncated Poisson distributions or a zero-modification of it.
Valero, J.; Pérez-Casany, M.; Ginebra, J. On poisson-stopped-sums that are mixed poisson. "Statistics and probability letters", 2013, vol. 83, núm. 8, p. 1830-1834.