Summary: A Note on Cumulants and M¨obius Inversion
Let be a probability space equipped with a -algebra A and a probability
measure . For any p [1, +) we denote the corresponding real Lp
space by
Lp
(). Let E be the intersection of these Lp
spaces. We work in this space, in
order for arbitrary moments to be well defined. We will define a collection of
multilinear forms ET
n , n 1, on this space E:
ET
n : En
R
(X1, . . . , Xn) ET
n [X1, . . . , Xn]
This construction is by induction on n. One lets ET
1 [X] = E[X], the usual
expectation. Then for n 2 one lets
ET
n [X1, . . . , Xn] = E[X1 · · · Xn] -

Source: Abdesselam, Abdelmalek - Department of Mathematics, University of Virginia

Collections: Mathematics