Background
I was contemplating the notion of subindependence. It states that for two random varaibles on a shared probability space that the characteristic function of their sum is equal to their the scalar product of their individual characteristic functions:
\[\phi_{X+Y} = \phi_{X}(t) \phi_{Y}(t).\]Multiple Variables
The first obvious generalization of this definition is to go to a collection of random variables ${X_1, \ldots, X_n }$:
\[\phi_{\{X_1 + X_2 + \cdots + X_n \}}(t) = \prod_{j=1}^n \phi_{X_{j}}(t)\]Choice of Operator
Instead of just considering the sum of random variables, we could generalize to a measurable function $\odot: \mathcal{X} \times \mathcal{Y} \mapsto \mathcal{Z}$.
\[\phi_{\{X_1 \odot X_2 \odot \cdots \odot X_n \}}(t) = \prod_{j=1}^n \phi_{X_{j}}(t).\]This has a familiarity to it. Like how Fourier transforms are related to sums of independent random variables or Mellin transforms under certain assumptions are related to products of independent random variables. I have not thought about this enough lately, but perhaps one can formalize this as a class of diagonalization decompositions in a function space.
Copula Theory
Also, I wonder how subindependence relates to copula theory. Sklar’s theorem gives us general conditions for mapping the marginal probabilities to their joint probabilities via a copula. So in a sense copulae specify how collections of random variables are dependent (or independent, in the case of the independence copula). Anyway, I intuit that there is a connection here.
Conclusions
Just sharing passerby impressions of the notion of independence. It seems like something that can readily be generalized, although I did not cover any mathematical or practical application.