  We prove that, for general test functions, the limiting behavior of the
linear statistic of an independent entry random matrix is determined only by
the first four moments of the entry distributions. This immediately generalizes
the known central limit theorem for independent entry matrices with complex
normal entries. We also establish two central limit theorems for matrices with
real normal entries, considering separately functions supported exclusively on
and exclusively away from the real line. In contrast to previously obtained
results in this area, we do not impose analyticity on test functions.
