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After many years of investigations in the  Theory of Random Matrices , we can say today that a very important and advanced result occupies the central place in this theory:  CENTral REsolvent Law (CENTRE-LAW) for the traces of analytic function of random matrices,  proved in 1975, in [15, pp. 278–324]. In the present paper we continue to consider this important problem of  Theory of Random Matrices (TRM) -  the  CENTRE-LAW  for the resolvent’s trace of a certain empirical covariance matrix    of dimension m n  which is used in almost all known estimators of  General Statistical Analysis(GSA).  At the end of this paper the reader can find the literature concerning  GSA: [1–46, GSA]. Here we follow the main procedures of  REFORM  method  (REsolvent, FORmula and Martingale)  and have shown as 30 years ago that  Central Limit Theorem for the traces of analytic function of random matrices  has an unbelievable property: it is asymptotically normal with convergence rate ( m n n ) −1/2  under G-condition  m n n  −1  &lt;1, where  n  is the number of independent observations    of a random vector    with covariance matrix   .

Thirty years of the Central Resolvent Law and three laws on the 1/ n expansion for resolvent of random matrices