CLT for linear spectral statistics of normalized sample covariance matrices with the dimension much larger than the sample size
Date of Issue2015
School of Physical and Mathematical Sciences
Let A = 1/√np(XT X−pIn) where X is a p×n matrix, consisting of independent and identically distributed (i.i.d.) real random variables Xij with mean zero and variance one. When p/n→∞, under fourth moment conditions a central limit theorem (CLT) for linear spectral statistics (LSS) of A defined by the eigenvalues is established. We also explore its applications in testing whether a population covariance matrix is an identity matrix.
© 2015 Bernoulli Society for Mathematical Statistics and Probability. This paper was published in Bernoulli and is made available as an electronic reprint (preprint) with permission of Bernoulli Society for Mathematical Statistics and Probability. The paper can be found at the following official DOI: [http://dx.doi.org/10.3150/14-BEJ599]. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law.