Cumulants in Noncommutative Probability Theory IV. De Finetti's Theorem and $L^p$-Inequalities

Abstract

In this paper we collect a few results about exchangeability systems in which crossing cumulants vanish, which we call noncrossing exchangeability systems. The main result is a free version of De Finetti's theorem, characterising amalgamated free products as noncrossing exchangeability systems which satisfy a so-called weak singleton condition. The main tool in the proof is an $L^p$-inequality with uniformly bounded constants for i.i.d. sequences in noncrossing exchangeability systems.

33 pages, AMS LaTeX; Brillinger's formula transferred to separate paper

References in corpus (2)

• Cumulants in noncommutative probability I. Noncommutative Exchangeability Systems
• Multilinear function series and transforms in free probability theory

Cited by in corpus (4)

• A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation
• Gaussian fluctuations of characters of symmetric groups and of Young diagrams
• De Finetti theorem on the CAR algebra
• On Lehner's `free' noncommutative analogue of De Finetti's theorem