A ROLE OF SINGLETONS IN QUANTUM CENTRAL LIMIT THEOREMS

ABSTRACT. A role of singletons in quantum central limit theorems is studied. A common feature of quantum central limit distribu· tions, the singleton condition which guarantees the symmetry of the limit distributions, is revisited in the category of discrete groups and monoids. Introducing a general notion of quantum independence, the singleton independence which include the singleton condition as an extremal case, we clarify the role of singletons and investigate the mechanism of arising non-symmetric limit distributions