Mod-$\phi$ convergence of Stirling distributions and limit theorems for zeros of their generating functions

We study mod-$\phi$ convergence of several probability distributions on the set of positive integers that involve Stirling numbers of both kinds and, as a consequence, derive various limit theorems for these distributions. We also derive closely related limit theorems for the distribution of zeros of the corresponding generating functions. For example, we identify the asymptotic distribution of zeros for the generating polynomial of the number of occupied boxes when $n$ balls are allocated equiprobably and independently among $\theta$ boxes in the regime when $\theta$ grows linearly with $n$.