  In the paper [25], written in collaboration with Gesine Reinert, we proved a
universality principle for the Gaussian Wiener chaos. In the present work, we
aim at providing an original example of application of this principle in the
framework of random matrix theory. More specifically, by combining the result
in [25] with some combinatorial estimates, we are able to prove
multi-dimensional central limit theorems for the spectral moments (of arbitrary
degrees) associated with random matrices with real-valued i.i.d. entries,
satisfying some appropriate moment conditions. Our approach has the advantage
of yielding, without extra effort, bounds over classes of smooth (i.e., thrice
differentiable) functions, and it allows to deal directly with discrete
distributions. As a further application of our estimates, we provide a new
"almost sure central limit theorem", involving logarithmic means of functions
of vectors of traces.
