What it means by "asymptotic normality" properties of a random matrix?

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I know that for the case of a random variable and a random vector, one can using (multivariate) density of normal distribution and concepts of convergence to define an asymptotic normality of a random variable (vector). What about if we have to address this property for a random matrix? What fields should I read into to know more about this subject?

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maybe the normality means the distribution of the eigenvalues of the matrix. here are some reference. Accessible Intro to Random Matrix Theory (RMT) hope it will help.