The Cramér condition for the Curie–Weiss model of SOC

We pursue the study of the Curie-Weiss model of self-organized criticality we designed in arXiv:1301.6911. We extend our results to more general interaction functions and we prove that, for a class of symmetric distributions satisfying a Cram\'er condition $(C)$ and some integrability hypothesis, the sum $S_{n}$ of the random variables behaves as in the typical critical generalized Ising Curie-Weiss model. The fluctuations are of order $n^{3/4}$ and the limiting law is $k \exp(-\lambda x^{4})\,dx$ where $k$ and $\lambda$ are suitable positive constants. In arXiv:1301.6911 we obtained these results only for distributions having an even density.