V. L. Girko
Thirty years of the Central Resolvent Law and three laws on the 1/n expansion for resolvent of random matrices
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].
Random Operators and Stochastic Equations, Walter de Gruyter
Print ISSN: 0926-6364
Volume: 11, 06/2003
Pages: 167 - 212
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