FREE NESTED CUMULANTS AND AN ANALOGUE OF A FORMULA OF
BRILLINGER
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.