The asymptotic distribution and Berry.Esseen bound of a new test for independence in high dimension with an application to stochastic optimization

Let X1, ., Xn be a random sample from a p-dimensional population distribution. Assume that c1n..p.c2n. for some positive constants c1, c2 and .. 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((logn)5/2/.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.