New integrated view at partial-sums distributions

Partial-sums discrete probability distributions occurred in description of many stochastic models. They were used also as a tool for creating new distributions, or as a link between known distributions. It is shown in this paper that every discrete distribution with only non-zero probabilities is a partial-sums distribution, and, moreover, that it has infinitely many parent distributions. The paper generalizes and unifies the concept of partial-sums distribution. Besides, it generalizes some risk models in insurance and revises some approaches to mathematical modelling in quantitative linguistics.