Quantifying Dependence Between Random Vectors: A New Index with Applications

This article proposes a new index for quantifying the degree of dependence between random vectors with the same dimension. The index takes values in [0, 1] and equals zero if and only if the random vectors are sub-independent. Unlike mere uncorrelatedness, sub-independence implies a stronger form of dependence while remaining strictly weaker than full independence. The proposed index is constructed via characteristic functions and admits a simplified representation in terms of moments. We establish its theoretical properties and derive a computationally efficient formula for the corresponding empirical measure. Furthermore, we investigate the asymptotic behavior and finite-sample performance of the estimator and demonstrate its practical utility through applications in machine learning, actuarial science, and renewal theory. A numerical comparison with Pearson correlation and squared distance correlation is also given under several dependence structures. We also give a real-data application to international stock and foreign exchange returns.