THE ASYMPTOTIC DISTRIBUTION AND BERRY-ESSEEN BOUND OF A NEW TEST FOR INDEPENDENCE IN HIGH DIMENSION WITH AN APPLICATION TO STOCHASTIC OPTIMIZATION

Let X-1 , . . . , X-n be a random sample from a p-dimensional population distribution. Assume that c(1)n(alpha) <= p <= c(2)n(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)/root 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.