Local Lift Dependence Scale.

We study the Radon-Nikodym derivative $L \coloneqq d\mathbb{P}(X,Y)/d(\mathbb{P}(X) \times \mathbb{P}(Y))$, between the joint distribution of a random vector $(X,Y)$ and the product measure generated by their marginal distributions, as a scale of local dependence, which we call Local Lift Dependence. This notion of local dependence is extended for when this derivative is not defined, contemplating a large class of distributions which are of interest in statistics. This extension is based on the Hausdorff dimension of the support of the singular part of $\mathbb{P}(X,Y)$. We argue that $L$ is more general and suitable to study variable dependence than other specific local dependence quantifiers and global dependence coefficients, as the Mutual Information, which is the expectation of $\log L$. An outline of how this dependence scale may be useful in statistics and topics for future researches are presented.