  Let $\mathbf{X}_1,...,\mathbf{X}_n$ be a random sample from a $p$-dimensional
population distribution. Assume that $c_1n^{\alpha}\leq p\leq c_2n^{\alpha}$
for some positive constants $c_1,c_2$ and $\alpha$. In this paper we introduce
a new statistic for testing independence of the $p$-variates of the population
and prove that the limiting distribution is the extreme distribution of type I
with a rate of convergence $O((\log n)^{5/2}/\sqrt{n})$. This is much faster
than $O(1/\log n)$, a typical convergence rate for this type of extreme
distribution. A simulation study and application to stochastic optimization are
discussed.
