Global Dependency Measure for Sets of Random Elements: “The Italian Problem” and Some Consequences

We suggest a detailed analysis of the classical independence/dependence properties for finite sets of random events or variables. All possible combinations of random elements are considered as a configuration obeying a hierarchical property. We define a function called a Dependency Measure taking values in the interval [0, 1] (0 corresponds to mutually independent sets; 1 corresponds to totally dependent sets) and serving as a global measure of the amount of dependency which is contained in the whole set of random elements. This leads to “The Italian Problem” about the existence of a probability model and a set of random elements with any prescribed independence/dependence structure. Some consequences and nonstandard illustrative examples are given. Related properties such as exchangeability and association are also discussed.