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CLT for linear spectral statistics of large-dimensional sample covariance matrices. (English) Zbl 1063.60022
Statisticians are confronted with the task of analyzing data with high-dimensional data. In the past, so-called dimension reduction schemes were used in dealing with it. But even it is possible to compute much of what is needed, there is a fundamental problem with the analytical tools used by statisticians. Their use relies on their asymptotic behavior as the number of samples increase. Some methods behave very poorly and some are even not applicable. Let $$B_n = (1/N)T^{1/2}_{n}X_{n}X^{*}_{n}T^{1/2}_{n}$$ where $$X_n =(X_{ij})$$ is $$n \times N$$ with i.i.d. complex standardized entries having finite fourth moment, and $$T^{1/2}_{n}$$ is a Hermitian square root of the nonnegative definite Hermitian matrix $$T_{n}$$. The limiting behavior, as $$n \rightarrow\infty$$ with $$n/N$$ approaching a positive constant, of functionals of the eigenvalues of $$B_n$$, where each is given equal weight, is studied. Due to the limiting behavior of the empirical spectral distribution of $$B_n$$, it is known that these linear spectral statistics converge a.s. to a nonrandom quantity. The authors show their rate of convergence to be $$1/n$$ by proving, after proper scaling, that they form a tight sequence. Moreover, they show if $$XX^{2}_{11}=0$$ and $$X| X_{11}| ^4 =2$$, or if $$X_{11}$$ and $$T_n$$ are real and $$XX^{4}_{11} =3$$, they are shown to have Gaussian limits. The main theorem can be viewed as an extension of results obtained by D. Jonsson [J. Multivariate Anal. 12, 1-38 (1982; Zbl 0491.62021)] where the entries of $$X_n$$ are Gaussian and $$T_n = I$$ and is consistent with central limit theorem results on linear statistics of eigenvalues of other classes of random matrices. The techniques and arguments used to prove the theorem have nothing in common with any of the tools used in other papers.

##### MSC:
 60F05 Central limit and other weak theorems 15B52 Random matrices (algebraic aspects) 62G30 Order statistics; empirical distribution functions 62H05 Characterization and structure theory for multivariate probability distributions; copulas
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