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 α ≤ p≤ c2n for some positive constants c1, c2 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((logn)/ √ n). This is much faster than O(1/ logn), a typical convergence rate for this type of extreme distribution. A simulation study and application to stochastic optimization are discussed.