Is there a proof for the Central Limit Theorem via some fixed point theorem?

Question

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?

Answer 1

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)