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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'.

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    $\begingroup$ I asked here because my professor (who is a renowned probablist) said he only saw this in a seminar, and he didn't know any books which describe this result. In light of this, I don't see why my question merits a downvote. Also, most measure theory books do not even mention probability. $\endgroup$
    – Exterior
    Commented Dec 30, 2014 at 17:42
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    $\begingroup$ @kaleidoscop Really? So what action is being considered when they speak of fixed points of orbits? Are you referring to the fact that the convolution of independent Gaussians is again a Gaussian? $\endgroup$
    – Yemon Choi
    Commented Dec 30, 2014 at 18:07
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    $\begingroup$ that's right I read too quickly, sorry. So an orbit is the class of all $\mu\star\nu$ for fixed $\mu$ and $\nu$ a probability measure. So what does it mean for the Gaussian distribution "to be the fixed point of each orbit"? $\endgroup$
    – kaleidoscop
    Commented Dec 30, 2014 at 20:50
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    $\begingroup$ @kaleidoscop I don't know what the precise meaning is, that's what I'm trying find out. Also, could you please remove your downvote? :) $\endgroup$
    – Exterior
    Commented Dec 30, 2014 at 20:58
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    $\begingroup$ there you go (I had to upvote, though... :) $\endgroup$
    – kaleidoscop
    Commented Dec 30, 2014 at 21:05

4 Answers 4

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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 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.

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Indeed, as J.C. said this has to do with the renormalization group (RG) which in the present context is a transformation $\mu\rightarrow \mu\ast\mu$ followed by rescaling by $\sqrt{2}$ to keep the variance the same. The "orbits" are the trajectories or sequences of iterates of a given probability measure by that RG transformation. The standard Gaussian is an attractive fixed point to which all these trajectories converge. This is one way to understand the central limit theorem. See this MO question for more info on this and in particular the paper by Anshelevich mentioned in the comment therein by Yemon Choi.

Also, one of the first references in this circle of ideas is the article "The renormalization group: A probabilistic view" by Jona-Lasinio.

Finally you can find more explanations about the RG in my answer to this MO question.

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I believe the following book by Yakov Sinai: "Probability Theory An Introductory Course" (Springer) uses such an argument to prove a version of the central limit theorem in chapter 15. UPDATE A better reference is the updated version of that book, Leonid Koralov & Yakov Sinai: "Theory of Probability and Random Processes, second edition" (Universitext).

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

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