  In this paper, we investigate a continuous family of notions of independence
which interpolates between the classical and free ones for non-commutative
random variables. These notions are related to the liberation process
introduced by D. Voiculescu. To each notion of independence correspond new
convolutions of probability measures, for which we establish formulae and of
which we compute simple examples. We prove that there exists no reasonable
analogue of classical and free cumulants associated to these notions of
independence.
