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On the convergence of the spectral empirical process of Wigner matrices. (English) Zbl 1101.60012
Summary: It is well known that the spectral distribution \(F_n\) of a Wigner matrix converges to Wigner’s semicircle law. We consider the empirical process indexed by a set of functions analytic on an open domain of the complex plane including the support of the semicircle law. Under fourth-moment conditions, we prove that this empirical process converges to a Gaussian process. Explicit formulae for the mean function and the covariance function of the limit process are provided.

MSC:
60F05 Central limit and other weak theorems
15B57 Hermitian, skew-Hermitian, and related matrices
82B31 Stochastic methods applied to problems in equilibrium statistical mechanics
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