Tests for the significance of a correlation matrix via ℓa-norms in high-dimensions

This paper presents a new method for evaluating the significance of correlations in high-dimensional contexts. The test statistic aggregates U-statistic based estimates of pairwise correlations raised to the $a$-th power, preserving invariance under location and scale transformations. We demonstrate that, under the null hypothesis, a collection of such statistics with distinct power parameters converges to a multivariate Gaussian distribution with identity covariance. This theoretical framework facilitates an efficient simultaneous testing procedure that incorporates multiple power parameters.