2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/216329Let $\mathbf{X}_1,...,\mathbf{X}_n$ be a random sample from a $p$-dimensional population distribution. Assume that $c_1n^α\leq p\leq c_2n^α$ for some positive constants $c_1,c_2$ and $α$. 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.Published in at http://dx.doi.org/10.1214/08-AAP527 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)Probability60F05 (Primary) 62F05 (Secondary)text