Quantifying Dependence Between Random Vectors: A New Index with Applications

TL;DR

This paper introduces a novel dependence index for random vectors, based on characteristic functions, which captures a stronger dependence than uncorrelatedness but weaker than full independence, with theoretical and practical implications.

Contribution

The paper presents a new dependence index for random vectors, including its theoretical properties, efficient computation, and applications across multiple fields.

Findings

Index ranges in [0,1], zero iff sub-independent

Derived a computationally efficient empirical measure

Demonstrated practical utility in machine learning, actuarial science, and renewal theory

Abstract

This article proposes a new index for quantifying the degree of dependence between random vectors. 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 of the estimator and demonstrate its practical utility through applications in machine learning, actuarial science, and renewal theory.