  We show central limit theorems (CLT) for the Stieltjes transforms or more
general analytic functions of symmetric matrices with independent heavy tailed
entries, including entries in the domain of attraction of $\alpha$-stable laws
and entries with moments exploding with the dimension, as in the adjacency
matrices of Erd\"os-R\'enyi graphs. For the second model, we also prove a
central limit theorem of the moments of its empirical eigenvalues distribution.
The limit laws are Gaussian, but unlike to the case of standard Wigner
matrices, the normalization is the one of the classical CLT for independent
random variables.
