2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/157455We study probability distributions of convergent random series of a special structure, called perpetuities. By giving a new argument, we prove that such distributions are of pure type: degenerate, absolutely continuous, or continuously singular. We further provide necessary and sufficient criteria for the finiteness of $p$-moments, $p>0$ as well as exponential moments. In particular, a formula for the abscissa of convergence of the moment generating function is provided. The results are illustrated with a number of examples at the end of the article.to appear in Journal of Theoretical ProbabilityProbability60E99; 60G50text