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In this paper, we investigate a continuous family of notions of independencewhich interpolates between the classical and free ones for non-commutativerandom variables. These notions are related to the liberation processintroduced by D. Voiculescu. To each notion of independence correspond newconvolutions of probability measures, for which we establish formulae and ofwhich we compute simple examples. We prove that there exists no reasonableanalogue of classical and free cumulants associated to these notions ofindependence.

A continuous semigroup of notions of independence between the classical and the free one