  The categorical Gini correlation proposed by Dang et al. is a dependence
measure to characterize independence between categorical and numerical
variables. The asymptotic distributions of the sample correlation under
dependence and independence have been established when the dimension of the
numerical variable is fixed. However, its asymptotic behavior for high
dimensional data has not been explored. In this paper, we develop the central
limit theorem for the Gini correlation in the more realistic setting where the
dimensionality of the numerical variable is diverging. We then construct a
powerful and consistent test for the $K$-sample problem based on the asymptotic
normality. The proposed test not only avoids computation burden but also gains
power over the permutation procedure. Simulation studies and real data
illustrations show that the proposed test is more competitive to existing
methods across a broad range of realistic situations, especially in unbalanced
cases.
