CLT for linear spectral statistics of normalized sample covariance matrices with the dimension much larger than the sample size

Let $\mathbf{A}=\frac{1}{\sqrt{np}}({\mathbf{X}}^T\mathbf{X}-p{\mathbf{I}}_n)$ where $\mathbf{X}$ is a $p\times n$ matrix, consisting of independent and identically distributed (i.i.d.) real random variables $X_{ij}$ with mean zero and variance one. When $p\mathrm{/}n\to \mathrm{\infty }$, under fourth moment conditions a central limit theorem (CLT) for linear spectral statistics (LSS) of $\mathbf{A}$ defined by the eigenvalues is established. We also explore its applications in testing whether a population covariance matrix is an identity matrix.