  In this paper, our objective is to present a constraining principle governing
the spectral properties of the sample covariance matrix. This principle
exhibits harmonious behavior across diverse limiting frameworks, eliminating
the need for constraints on the rates of dimension $p$ and sample size $n$, as
long as they both tend to infinity. We accomplish this by employing a suitable
normalization technique on the original sample covariance matrix. Following
this, we establish a harmonic central limit theorem for linear spectral
statistics within this expansive framework. This achievement effectively
eliminates the necessity for a bounded spectral norm on the population
covariance matrix and relaxes constraints on the rates of dimension $p$ and
sample size $n$, thereby significantly broadening the applicability of these
results in the field of high-dimensional statistics. We illustrate the power of
the established results by considering the test for covariance structure under
high dimensionality, freeing both $p$ and $n$.
