[sobellian]

IIRC there's a video by 3b1b that talks about that, and it is important that gaussians are closed under convolution.

[gowld]

That makes it an equilibrium point in function space, but the other half is why it's an a global attractor.

[pfortuny]

There must be a contractive nature in "passing to the limit". And then Brower's fixed point theorem.

(I know it is very easy to do "maths" this way).