Due to massive numbers of uncoordinated devices present in wireless networks for the Internet of Things (IoT), interference is a key challenge. There is evidence both from experiments and analysis of statistical models that the uncoordi-nated nature of channel access leads to non-Gaussian statistics for the interference. A particularly attractive model in this scenario is the additive vector α-stable noise channel. In this paper, we study the capacity of this channel with fractional moment constraints. In particular, we establish well-posedness of the optimization problem for the capacity. We also study convergence of the capacity loss due to an additional constraint where input probability measures are concentrated on spherical shells, in addition to the fractional moment constraints.