has anybody heard of Free Probability?

it's a version of probability which uses a complicated formulation of operator theory to define the sum of two "random variables" that are independent from one another in a new sense. the basic idea is that if you have two quantum mechanical operators, their eigenvalues represent outcomes of experiments, two such operators might be non-commutative (as is common in quantum mechanics) but "independent" in some sense. the classical probability scenario is basically where the operators are diagonal, or the commutative case. there is a version of independence that is on the other extreme, where two operators are totally non-commutative in the most extreme case (their eigenvalues are in eigenspaces that are uniformly randomly distributed relative to one another).

this leads to a way to sum independent copies of probability measures to produce new ones, but they don't follow the usual results from probability. a key example is in random matrix theory, where large random matrices have eigenvalues that are predicted by these free independent sums