I was reading a proof of the claim that the convolution of two Gaussians with arbitrary mean and variance results in another Gaussian. It can be found here.
At the end of the proof, which I studied in detail, the author claims "the preservation of the normalization when convolving PDFs ... is a special case rather than being true in general."
I take this to mean that the result of the convolution (of 2 normal RV's) being a normal (Gaussian) distribution is somehow not general. But this is certainly not what I understood from the proof - did the author not just prove the result in its full generality (convolution of 2 Gaussian is Gaussian)? Perhaps I missed something in the proof?
From what you have stated, no, that is not what he meant. He is saying that the sum of two normal distributions is again normal. However this is not always the case. That is what he is saying. He is saying that this is a coincidence/special for the sum of normals. He is saying that the sum of two $iid$ random variables following some distribution $F$ is not again $F$. For example, the sum of two $iid$ exponential distributions is not another exponential. If the rates are the same, then it is however a gamma distribution.