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Sample covariance matrices are widely used in multivariate statisticalanalysis. The central limit theorems (CLT's) for linear spectral statistics ofhigh-dimensional non-centered sample covariance matrices have receivedconsiderable attention in random matrix theory and have been applied to manyhigh-dimensional statistical problems. However, known population mean vectorsare assumed for non-centered sample covariance matrices, some of which evenassume Gaussian-like moment conditions. In fact, there are still another twomost frequently used sample covariance matrices: the MLE (by subtracting thesample mean vector from each sample vector) and the unbiased sample covariancematrix (by changing the denominator $n$ as $N=n-1$ in the MLE) withoutdepending on unknown population mean vectors. In this paper, we not onlyestablish new CLT's for non-centered sample covariance matrices withoutGaussian-like moment conditions but also characterize the non-negligibledifferences among the CLT's for the three classes of high-dimensional samplecovariance matrices by establishing a {\em substitution principle}: substitutethe {\em adjusted} sample size $N=n-1$ for the actual sample size $n$ in themajor centering term of the new CLT's so as to obtain the CLT of the unbiasedsample covariance matrices. Moreover, it is found that the difference betweenthe CLT's for the MLE and unbiased sample covariance matrix is non-negligiblein the major centering term although the two sample covariance matrices onlyhave differences $n$ and $n-1$ on the dominator. The new results are applied totwo testing problems for high-dimensional data.

Substitution principle for CLT of linear spectral statistics of high-dimensional sample covariance matrices with applications to hypothesis testing