  Here we extend the algebro-geometric approach to free probability, started
in~\cite{FMcK4,F14}, to general (non)-commutative probability theories. We show
that any universal convolution product of moments of independent
(non)-commutative random variables defined on a graded connected dual
semi-group is given by a pro-unipotent group scheme. We show that
moment-cumulant formulae have a natural interpretation within the theory of
homogeneous Lie groups, which we generalise for the present purpose, and are
given by the log and exp map, respectively. Finally, we briefly discuss the
universal role of the shuffle Hopf algebra.
