Title: Asymptotic power of Rao's score test for independence in high dimensions,
Abstract: Let ${\bf R}$ be the Pearson correlation matrix of $m$ normal random
variables. The Rao's score test for the independence hypothesis $H_0 : {\bf R}
= {\bf I}_m$, where ${\bf I}_m$ is the identity matrix of dimension $m$, was
first considered by Schott (2005) in the high dimensional setting. In this
paper, we study the asymptotic minimax power function of this test, under an
asymptotic regime in which both $m$ and the sample size $n$ tend to infinity
with the ratio $m/n$ upper bounded by a constant. In particular, our result
implies that the Rao's score test is rate-optimal for detecting the dependency
signal $\|{\bf R} - {\bf I}_m\|_F$ of order $\sqrt{m/n}$, where $\|\cdot\|_F$
is the matrix Frobenius norm.