In this paper, we consider the limit properties of the largest entries of sample covariance matrices and the sample correlation matrices. In order to make the statistics based on the largest entries of the sample covariance matrices and the sample correlation matrices more applicable in high-dimensional tests, the identically distributed assumption of population is removed. Under some moment’s assumption of the underlying distribution, we obtain that the almost surely limit and asymptotical distribution of the extreme statistics as both the dimension
p
and sample size
n
tend to infinity.
Substitution principle for CLT of linear spectral statistics of high-dimensional sample covariance matrices with applications to hypothesis testing
