Statistics and Data Science Seminar
Han Xiao
University of Chicago
Almost Sure Limit of the Smallest Eigenvalue of Some Sample Correlation Matrices
Abstract: Let $X^{(n)}=(X_{ij})$ be a $p \times n$ data matrix, where the $n$ columns form a random sample of size $n$ from a certain $p$-dimensional
distribution. Let $R^{(n)}=(\rho_{ij})$ be the $p \times p$ sample correlation coefficient matrix of $X^{(n)}$; and $S^{(n)} =
(1/n)X^{(n)}\left(X^{(n)}\right)^{\ast}-\bar{X}\bar{X}^{\ast}$ be the sample covariance matrix of $X^{(n)}$, where $\bar{X}$ is the mean vector of the
$n$ observations. Assuming that $X_{ij}$'s are independent and identically distributed with finite fourth moment, we show that the smallest eigenvalue
of $R^{(n)}$ converges almost surely to the limit $(1-\sqrt{c}\,)^2$ as $n \rightarrow \infty$ and $p/n \rightarrow c \in (0\,,\,\infty)$. We
accomplish this by showing that the smallest eigenvalue of $S^{(n)}$ converges almost surely to $(1-\sqrt{c}\,)^2$.
Wednesday February 17, 2010 at 3:00 PM in SEO 636