  The sequence of so-called signature moments describes the laws of many
stochastic processes in analogy with how the sequence of moments describes the
laws of vector-valued random variables. However, even for vector-valued random
variables, the sequence of cumulants is much better suited for many tasks than
the sequence of moments. This motivates us to study so-called signature
cumulants. To do so, we develop an elementary combinatorial approach and show
that in the same way that cumulants relate to the lattice of partitions,
signature cumulants relate to the lattice of so-called "ordered partitions". We
use this to give a new characterisation of independence of multivariate
stochastic processes; finally we construct a family of unbiased
minimum-variance estimators of signature cumulants.
