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

Linear Statistics of Non-Hermitian Matrices Matching the Real or Complex Ginibre Ensemble to Four Moments