Gaussian distributions as fixed points in Some distribution space

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.

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.

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).