In an irreducible stochastic game, no single player can prevent the stochastic process on states from being irreducible, so the other players can ensure that the current state has little effect on events in the distant future. This paper introduces stochastic games with imperfect public signals, and provides a sufficient condition for the folk theorem when the game is irreducible, thus generalizing the folk theorems of Dutta (1995) and Maskin (1994) . To prove this theorem, the paper extends

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... concept of self-generation (Abreu, Pearce, and Stachetti (1990)) to "return generation," which explicitly tracks actions and incentives until the next time the state returns to its current value, and asks that players not wish to deviate given the way their continuation payoffs from the time of this return depend on the public signals that have been observed.