On de Finetti-Type Theorems

The investigation of distributional symmetries was initiated by de Finetti’s celebrated theorem, which shows that any finite joint-distribution of sequences of two-point valued exchangeable random variables is obtained by randomization of the binomial distribution. This result has since found several generalizations both in classical and noncommutative settings. In this paper, we discuss a series of recent results that extend de Finetti’s theorem to various non-commutative models, including Bolean, monotone, CAR algebra as well as \(\mathbb {Z}_2\) -graded \(C^*\) -algebras.