Extreme eigenvalues of principal minors of random matrix with moment conditions

Abstract

Let ${\mathbf{x}}_1,\cdots ,{\mathbf{x}}_n$ be a random sample of size $n$ from a $p$-dimensional population distribution, where $p=p(n)\to \infty$. Consider a symmetric matrix $W=X^{\top }X$ with parameters $n$ and $p$, where $X=({\mathbf{x}}_1,\cdots ,{\mathbf{x}}_n{)}^{\top }$. In this paper, motivated by model selection theory in high-dimensional statistics, we mainly investigate the asymptotic behavior of the eigenvalues of the principal minors of the random matrix $W$. For the Gaussian case, under a simple condition that $m=o(n/\log p)$, we obtain the asymptotic results on maxima and minima of the eigenvalues of all $m\times m$ principal minors of $W$. We also extend our results to general distributions with some moment conditions. Moreover, we gain the asymptotic results of the extreme eigenvalues of the principal minors in the case of the real Wigner matrix. Finally, similar results for the maxima and minima of the eigenvalues of all the principal minors with a size smaller than or equal to $m$ are also given.