Distance correlation test for high-dimensional independence

In this paper, a new self-normalized and scale invariant statistic Tn, which is based on distance correlations, is developed for testing mutual independence of a high-dimensional random vector. The asymptotic normality of the statistic is established under mild moment conditions by assuming both the dimension p of the vector and the sample size n grow to infinity. In particular, the test procedure has the consistency against sparse alternatives where the dependence can be nonlinear and non-monotonic. Technically, the asymptotic normality of the test statistic is built upon martingale decomposition and novel moment method with appropriate combinatorics.