CUMULANTS-IN-NONCOMMUTATIVE-PROBABILITY-THEORY-IV-NONCROSSING-CUMULANTS-DE-FINETTI’S-THEOREM,Lp-INEQUALITIES-AND-BRILLINGER’S-FORMULA.pdfVIP

CUMULANTS IN NONCOMMUTATIVE PROBABILITY THEORY IV. NONCROSSING CUMULANTS: DE FINETTI’S THEOREM, p L -INEQUALITIES AND BRILLINGER’S FORMULA FRANZ LEHNER Abstract. 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 con- ditionally i.i.d. with respect to some σ-algebra. In this paper we prove a “free” noncom- mutative 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 p noncommutative L -inequality for i.i.d. sums of centered noncommutative random vari- ables 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. Date : May 30, 2006. 1991 Mathematics Subject Classification. Primary 46L53, Secondary 05A18. Key words and phrases. Cumulants, partition lattice, M¨obius inversion, free probability, noncrossing partitions, noncommutative probability. Supported by the Austrian Science Fund (FWF) Project ›R2-MAT. 1 2