We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
Thirty years of the Central Resolvent Law and three laws on the 1/n expansion for resolvent of random matrices.
- Authors
Girko, V. L.
- Abstract
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 &Rcirc;m[subn] of dimension m[subn] 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 functional random matrices has an unbelievable property: it is asymptotically normal with convergence rate (m[subn]n)[sup-1/2] under G-condition m[subn]n-1 < 1, where n is the number of independent observations x[sup→(1)]m[subn],...,x[sup→(n)]m[subn] of a random vector ξ[sup→]m[subn] with covariance matrix Rm[subn]. 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 anaiytic function of random matrices, we describe very succinctly the main features of the proof of the CENTRE-law. As in the previous...
- Subjects
THEORY; RANDOM matrices; MATRICES (Mathematics); MATHEMATICAL functions; MATHEMATICAL statistics
- Publication
Random Operators & Stochastic Equations, 2003, Vol 11, Issue 2, p167
- ISSN
0926-6364
- Publication type
Article
- DOI
10.1163/156939703322386913