On quantum distributional symmetries for *-random variables

In this talk, we briefly review the distributional symmetries for *-random variables, which are defined by coactions of corepresentations of quantum groups. We classify all de Finetti type theorems for classical independence and free independence by studying vanishing conditions on the classical and free cumulants.  Examples for our de Finetti type theorems and approximation results in the spirit of Diaconis and Freedman are also provided.