A Brunnian Theorem for Finite Families of Random Variables

In 2014, during a study on the connectivity structures of quantum entanglement, I specifically introduced the notion of ‘‘the connectivity structure of a family of random variables’’ – a structure that expresses the dependency relations between the variables in question – and I stated the following proposition, which can be described as Brunnian in reference to Hermann Brunn’s work on links (1892) : “Every finite connectivity structure is that of a family of random variables”. At the time, however, I neglected to write down the proof of this assertion, merely providing an intuitive idea of it. The purpose of this article is to present such a proof.