The Cram\'er 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 $n^{3\mathrm{/}4}$ 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.