  The theory of cumulants is revisited in the "Rota way", that is, by following
a combinatorial Hopf algebra approach. Monotone, free, and boolean cumulants
are considered as infinitesimal characters over a particular combinatorial Hopf
algebra. The latter is neither commutative nor cocommutative, and has an
underlying unshuffle bialgebra structure which gives rise to a shuffle product
on its graded dual. The moment-cumulant relations are encoded in terms of
shuffle and half-shuffle exponentials. It is then shown how to express
concisely monotone, free, and boolean cumulants in terms of each other using
the pre-Lie Magnus expansion together with shuffle and half-shuffle logarithms.
