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

Abstract

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 $p/n\to \infty$, 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.

Published at http://dx.doi.org/10.3150/14-BEJ599 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (2)

• High-dimensional classification using features annealed independence rules
• Convergence of the largest eigenvalue of normalized sample covariance matrices when p and n both tend to infinity with their ratio converging to zero

Cited by in corpus (1)

• Asymptotic joint distribution of extreme eigenvalues and trace of large sample covariance matrix in a generalized spiked population model