Singular vector distribution of sample covariance matrices

Abstract

We consider a class of sample covariance matrices of the form $Q=TX{X}^{\ast }{T}^{\ast },$ where $X=(x_{ij})$ is an $M\times N$ rectangular matrix consisting of i.i.d entries and $T$ is a deterministic matrix satisfying $T^{\ast }T$ is diagonal. Assuming $M$ is comparable to $N$, we prove that the distribution of the components of the singular vectors close to the edge singular values agrees with that of Gaussian ensembles provided the first two moments of $x_{ij}$ coincide with the Gaussian random variables. For the singular vectors associated with the bulk singular values, the same conclusion holds if the first four moments of $x_{ij}$ match with those of Gaussian random variables. Similar results have been proved for Wigner matrices by Knowles and Yin.

References in corpus (4)

• Near-optimal bounds for phase synchronization
• On asymptotics of eigenvectors of large sample covariance matrix
• A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices
• Local circular law for the product of a deterministic matrix with a random matrix

Cited by in corpus (2)

• A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices
• The Asymptotic Distribution of Modularity in Weighted Signed Networks