Random models on regularity-integrability structures

We prove a convergence result for a large class of random models that encompasses the case of the BPHZ models used in the study of singular stochastic PDEs. We introduce for that purpose a useful variation on the notion of regularity structure called a regularity-integrability structure. It allows to deal in a single elementary setting with models on a usual regularity structure and their first order Malliavin derivative.