Widely used convolutions and deconvolutions techniques traditionally rely on the assumption of independence, an assumption often criticized as being very strong. We observe that independence is, in fact, not necessary for the convolution theorem to hold. Instead, a much weaker notion, known as subindependence, is the appropriate necessary and sufficient condition. We motivate the usefulness of the subindependence concept by showing that is arguably as weak as a conditional mean assumption. We also provide an equivalent definition of subindependence that does not involve Fourier transforms and devise a constructive method to generate pairs of subindependent random variables.