The process of crystallization of polyrners may be modelled as a spatially structured counting process, whose intensity kernel depends upon the available free volume. Due to the effect of impingement, an explicit expression for the stochastic occupied volume is difficult to obtain, while derivations of its expected value have been provided since the pioneering work of Kolmogorov and Avrami, in use of chemical engineers. In this paper we provide two theorems which, by the use of stochastic geometry, ensure local and global convergence of the stochastic growth process to its expected value, justifing so the use of deterministic models to predict the results of real experiments.