We study stochastic games with a large number of players, where players are coupled via their payoff functions. A standard solution concept for stochastic games is Markov perfect equilibrium (MPE). In MPE, each player's strategy is a function of its own state as well as the state of other players. This makes MPE computationally prohibitive as the number of players becomes large. An approximate solution concept called oblivious equilibrium (OE) was introduced by Weintraub et al., where each player's decision depends only on its own state and the "long-run average" state of other players. This makes OE computationally more tractable than MPE. It was shown that under a set of assumptions, as the number of players becomes large, OE closely approximates MPE. However, these assumptions require the computation of OE and verifying that the resulting stationary distribution satisfies a certain light-tail condition. In this paper, we derive exogenous conditions on the state dynamics and the payoff function under which the light-tail condition holds. A key condition is that the agents' payoffs are concave in their own state and actions. These exogenous conditions enable us to characterize a family of stochastic games in which OE is a good approximation for MPE.