Hmm, you know, I think you have a point. Clearly any CLT solution must be a fixed point of convolution (as the Gaussian is). The other property ("Gaussian is

Hmm, you know, I think you have a point. Clearly any CLT solution must be a fixed point of convolution (as the Gaussian is). The other property ("Gaussian is its own Fourier transform") seemed important, but I'm unable to formalize the argument, and don't think that "fixed point of convolution" requires it. Thanks for keeping me honest. Unless I'm missing something, your observation would render the closing statement in the GP comment ("because the Gaussian is its own Fourier transform") completely irrelevant.