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

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