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



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> 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."
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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.



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> Where can I find a (preferably grad-level) reference for these ideas?
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**Clarification:** I am not asking about the fact 'the convolution of independent Gaussians is Gaussian'.



Gaussian distributions as fixed points in Some distribution space
Not sure if this is what you want, but orbits in spaces of probability distributions can be thought of as simple cases of **renormalization group flows** in statistical mechanics, see e.g. the discussion in [this paper of Calvo et al](http://link.springer.com/article/10.1007%2Fs10955-010-0065-y#page-1) and its references, particularly the book of Gnedenko and Kolmogorv, "Limit Distributions for Sums of Independent Random Variables". There's also [an article by Li and Sinai covering similar ground](http://link.springer.com/article/10.1007%2Fs10955-014-1098-4).


