TITLE: Gaussian distributions as fixed points in Some distribution space
QUESTION [10 upvotes]: 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'.

REPLY [2 votes]: The introductory example in this video: https://www.youtube.com/watch?v=lTIchf0V9qo actually addresses this.