Recently, Jacod, Kowalski and Nikeghbali introduced a new kind of  convergence called mod-gaussian.  It extends concepts behind the  central limit theorem.  They applied it to various situations.  Here  I  will give an extension and show how it is related to local limit  theorems.  The conditions we need are related to Gnedenko's  condition.  We think that this kind of convergence allows for a lot of  applications, e.g. number theory.  This is joint work with E. Kowalski