  We present an approach to classical definitions and results on
cumulant--moment relations and Wick polynomials based on extensive use of
convolution products of linear functionals on a coalgebra. This allows, in
particular, to understand the construction of Wick polynomials as the result of
a Hopf algebra deformation under the action of linear automorphisms induced by
multivariate moments associated to an arbitrary family of random variables with
moments of all orders. We also generalise the notion of deformed product in
order to discuss how these ideas appear in the recent theory of regularity
structures.
