Most introductory statistical mechanics books state that in the thermodynamic limit, ensemble averages go towards the value that corresponds to the most probable state.
They justify this statement using an example of a non-interacting system where the spin/energy of each molecule/particle/spin are statistically identical and independent. The probability distribution for the sping/energy in question goes towards a Gaussian distribution according to the Law of Large Numbers (LLN) and the Central Limit Theorem (CLM).
In most thermodynamic systems, molecules/particles/spins interact and are not statistically independent.
For systems with interacting molecules/spins, when can we still expect the ensemble average to correspond to the most probable state in the thermodynamic limit?