is there a proof of the central limit theorem via max entropy

#question #math

• the gaussian distribution is the max entropy distribution subject to a constraint on the variance

• intuitively, the dynamics that you get from continuing to add a random variable to itself should be increasing the entropy

• is there some argument that says that, this must always lead to convergence to max entropy distribution?

• another observation is in the setting of glauber dynamics, perhaps metropolis-hastings more generally, one also wants to prove convergence of to a max entropy distribution subject to an energy constraint...

  • is this related at all?