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 <a href="http://mathoverflow.net/questions/182752/central-limit-theorem-via-maximal-entropy">this MO question</a> 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 <a href="http://isites.harvard.edu/fs/docs/icb.topic1473685.files/Week%206/RG.pdf">"The renormalization group: A probabilistic view"</a> by Jona-Lasinio. 

Finally you can find more explanations about the RG in my answer to <a href="http://mathoverflow.net/questions/62770/what-mathematical-treatment-is-there-on-the-renormalization-group-flow-in-a-spac">this MO question</a>.