When we study correlation or association in real data, do we always (implicitly) assume a finite second order moment for any hypothetical population distribution? If we do not assume this, what statistical measure may we use?
I am not sure if it makes sense to study correlation (or association in general) without assuming a finite second order moment for a generating probabilistic model. In the generalized version of pearson correlation (wikipedia general correlation), the norms in the denominator of the expression should implicitly assume a finite second order moment.
I ask this question simply to keep track of what we do and do not assume when we look into association in datasets.
I think that association does not make sense without assuming a finite second order moment for any population distribution. However, maybe there is a nonparametric measure that I am not familiar with. Since both Spearman's and Kendall's correlation coefficients can be expressed in the form in 1, I think this implicit assumption of a finite second order moment applies to these two association measures also. Any insights would be greatly appreciated!
This stackexchange question seems relavent: PCA without finite second moment. However, as far as I can tell, it does not definitively answer the question as I understand it.