In this dissertation we present new results on the classification of limit distributions of random geometric processes. In particular, that develop on the work of Penrose and Wade, who were the first to document the phenomenon of infinite divisibility in the case of a particular (uniform) distribution. In this dissertation we put forth not only new results, but a new method of obtaining results through analyzing the sequence of moments produced by random variables. Additionally we have new results in cycle decomposition of the related Dickman-Goncharov distribution. We present a novel proof of the distribution of the three highest order cycles in a random partition.