  In this note, we characterize the limiting functions in mod-Gausssian
convergence; our approach sheds a new light on the nature of mod-Gaussian
convergence as well. Our results in fact more generally apply to mod-*
convergence, where * stands for any family of probability distributions whose
Fourier transforms do not vanish. We moreover provide new examples, including
two new examples of (restricted) mod-Cauchy convergence from arithmetics
related to Dedekind sums and the linking number of modular geodesics.
