The capacity of average power constrained additive non-Gaussian noise channels

It is well known that a Gaussian input achieves the capacity of the linear additive white Gaussian noise channel under the average power constraint. While the continuity of an optimal input of a continuous channel might be expected, the Gaussian noise presents the only such scenario. In fact, we study in this paper the capacity-achieving inputs for the linear additive noise channel where the noise is not necessarily Gaussian. We impose an average power constraint on the input and prove that, except for a Gaussian channel, the optimal input is of a discrete nature.