This paper studies infinite-horizon stochastic games in which players observe actions and noisy public information about a hidden state each period. We find a general condition under which the feasible and individually rational payoff set is invariant to the initial prior about the state, when players are patient. This result ensures that players can punish or reward the opponents via continuation payoffs in a flexible way. Then we prove the folk theorem, assuming that public randomization is available. The proof is constructive, and uses the idea of random blocks to design an effective punishment mechanism.