Cumulants in Noncommutative Probability Theory Iv. Noncrossing Cumulants: De Finetti’s Theorem, L-inequalities and Brillinger’s Formula

In this paper we collect a few results about exchangeability systems in which crossing cumulants vanish, which we call noncrossing exchangeability systems. De Finetti’s theorem states that any exchangeable sequence of random variables is conditionally i.i.d. with respect to some σ-algebra. In this paper we prove a “free” noncommutative analog of this theorem, namely we show that any noncrossing exchangeability system with a faithful state which satisfies a so called weak singleton condition, can be embedded into a free product with amalgamation over a certain subalgebra. Vanishing of crossing cumulants can be verified by checking a certain weak freeness condition. The proof follows the classical proof of De Finetti’s theorem, the main technical tool being a noncommutative L-inequality for i.i.d. sums of centered noncommutative random variables in noncrossing exchangeability systems. In the second part of the paper we establish a free analogue of Brillinger’s formula, which expresses free cumulants in terms of amalgamated free cumulants.