Sums of independent random variables and the generalized Dickman laws

The probabilistic Dickman law, defined by the known Dickman function, and its generalized versions are considered. In the paper, we obtain a general criterion of the weak convergence to these laws for the distributions of sums of independent non-negative random variables within the series scheme in the classical setting. Moreover, we get a special criterion of the convergence for the case when the summing random variables have finite expectations.