TITLE: Is there a proof for the Central Limit Theorem via some fixed point theorem?
QUESTION [8 upvotes]: This question arose in my mind when I learned that the Gaussian is a fixed point for the Fourier transform. On the other hand, in e.g. the Banach fixed point theorem we have convergence to a fixed point.
So are these two things (CLT and Gaussian being a fixed point of the Fourier transform) somehow connected?

REPLY [6 votes]: If I understand your question correctly, you may want to look at Stein's method (which in particular yields a way to prove the Central Limit Theorem).
(cf. also Wikipedia's entry on Stein's method)