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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 33, Fasc. 2,
pages 327 - 339
 

FREE NESTED CUMULANTS AND AN ANALOGUE OF A FORMULA OF BRILLINGER

Franz Lehner

Abstract: We prove a free analogue of Brillinger’s formula (sometimes called “law of total cumulance”) which expresses classical cumulants in terms of conditioned cumulants. As expected, the formula is obtained by replacing the lattice of set partitions by the lattice of noncrossing set partitions and using and an appropriate notion of noncommutative nested products. As an application we reprove a characterization of freeness due to Nica, Shlyakhtenko, and Speicher by Möbius inversion techniques, without recourse to the Fock space model for free random variables.

2000 AMS Mathematics Subject Classification: Primary: 46L54; Secondary: 05A18.

Keywords and phrases: Multivariate free cumulants, conditioned cumulants, Brillinger’s formula.

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