Asymptotic dependency structure of multiple signals: Asymptotic equipartition property for diagrams of probability spaces

We formalize the notion of the dependency structure of a collection of multiple signals, relevant from the perspective of information theory, artificial intelligence, neuroscience, complex systems and other related fields. We model multiple signals by commutative diagrams of probability spaces with measure-preserving maps between some of them. We introduce the asymptotic entropy (pseudo-)distance between diagrams, expressing how much two diagrams differ from an information-processing perspective. If the distance vanishes, we say that two diagrams are asymptotically equivalent. In this context, we prove an asymptotic equipartition property: any sequence of tensor powers of a diagram is asymptotically equivalent to a sequence of homogeneous diagrams. This sequence of homogeneous diagrams expresses the relevant dependency structure.