caa a22 4500 378866494 CHVBK 20180305123359.0 cr unu---uuuuu 161128e20030601xx s 000 0 eng 10.1515/156939703322386913 doi (NATIONALLICENCE)gruyter-10.1515/156939703322386913 Girko V. L. Department of Probability and Statistics, Michigan State University, East Lansing, Michigan 48824 Thirty years of the Central Resolvent Law and three laws on the 1/n expansion for resolvent of random matrices [Elektronische Daten] [V. L. Girko] 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 mn 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 (mnn)−1/2 under G-condition mnn −1 <1, where n is the number of independent observations of a random vector with covariance matrix . We want to emphasize that all known publications concerning the problem of estimation of functions of many parameters deal only with improvements of estimators. See, for example, jackknife and bootstrap methods. Only in [1-46,GSA] it was for the first time, shown that there exist in this analysis consistent estimators of some functions under the G-condition. Therefore, we can develop mathematical statistics under G-condition without any new restrictions for observations and statistical models. In the following sections we present a review of the main steps of the proof of the main assertions about Central Resolvent Law for the traces of analytic function of random matrices. We describe very succinctly the main features of the proof of the CENTRE-law. As in the previous papers, we will focus mainly on the limit theorem for random determinants. The proof is quite similar to the one proved in [15, pp. 278-324]. Copyright 2003, Walter de Gruyter Random Operators and Stochastic Equations Walter de Gruyter 11/2(2003-06-01), 167-212 0926-6364 11:2<167 2003 11 rose https://doi.org/10.1515/156939703322386913 text/html Onlinezugriff via DOI 1 research article jats NATIONALLICENCE 856 40 https://doi.org/10.1515/156939703322386913 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Girko V. L. Department of Probability and Statistics, Michigan State University, East Lansing, Michigan 48824 NATIONALLICENCE 773 0- Random Operators and Stochastic Equations Walter de Gruyter 11/2(2003-06-01), 167-212 0926-6364 11:2<167 2003 11 rose CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-gruyter