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

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

2004-09-02

arxiv.org/abs/math/0409025v1

Mathematics

Operator Algebras

31 pages, AMS LaTeX

Scientific paper

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. 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.

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