Convolution without independence

Widely used convolution and deconvolution techniques traditionally rely on independence assumptions, often criticized as being strong. We observe that the convolution theorem actually holds under a weaker assumption, known as subindependence. We show that this notion is arguably as weak as a conditional mean assumption. We report various simple characterizations of subindependence and devise constructive methods to generate subindependent random variables. We extend subindependence to multivariate settings and propose the new concepts of conditional and mean subindependence, relevant to measurement error problems. We finally introduce three tests of subindependence based on characteristic functions, generalized method of moments and randomization, respectively.