If () and () denote the ordinary moment and the ordinary cumulant corresponding to the indices in

If μ(ϒ) and κ(ϒ) denote the ordinary moment and the ordinary cumulant corresponding to the indices in ϒ = {i1,…,tp}, show that

μ (ϒ) = κ (ϒ) + ∑

{υ1,υ2}

κ (υ1)μ (υ2)

where {υ1,υ2} is a partition of ϒ into two non-empty blocks and the sum extends over all partitions such that i1 ∊ υ1. What purpose does the condition i1 ∊ υ1 serve? Show that this result generalizes the univariate identity in Exercise 2.3.