Full classification of de Finetti type theorems for *-random variables in classical and free probability

Abstract: Classical distributional symmetries can be described as invariance under the actions of semigroups (or groups) of matrix structures, and subsequently under the coactions of continuous functions on the matrix semigroups (or groups) generated by entry functions. By considering noncommutative entry functions on matrix structures, Woronowicz introduced corepresentations of compact quantum groups, namely Woronowicz's $C^*$-algebras (also known as compact matrix pseudogroups). We demonstrate that every nontrivial finite sequence of random variables admits a maximal distributional symmetry determined by a Woronowicz $C^*$-algebra. This establishes a probabilistic framework for classifying compact quantum groups. Furthermore, we classify all de Finetti-type theorems for *-random variables that are invariant under distributional symmetries arising from compact matrix quantum groups in both classical and free probability settings. Our results show that only finitely many types of de Finetti theorems exist in these contexts, and the associated categories of (quantum) groups are the easy (quantum) groups introduced by Banica and Speicher.