I'm taking a course on topology and probabily. Today, the professor remarked something along the lines of:

If you look at the space of probability distributions with $0$ mean and variance $1$, equipped with convolution, then the Gaussian distribution is characterized as the fixed point of each orbit."

He also said this was a nice way to appreciate the importance of the gaussian distribution, and to gain insight for the central limit theorem.

I asked for references on this point of view, but he said it's not standard and recalled only hearing about it in some seminar forty years ago.

Where can I find a (preferably grad-level) reference for these ideas?

**Clarification:** I am not asking about the fact 'the convolution of independent Gaussians is Gaussian'.