We derive statistical limit theorems for sample-based approximations of infinite-horizon discounted stochastic optimal control problems in discrete time. Our first result is a functional central limit theorem for the sample-based value function under a uniqueness-type condition on population optimal policies. The limiting law is a mean-zero Gaussian process characterized by a linear fixed point equation that resembles a dynamic programming principle. We compare these asymptotics with those obtained from sample-based policy optimization and illustrate that their limiting variances can be different. We also derive a limit theorem for models with nonunique optimal policies, where the limiting law may be non-Gaussian. Applications to inventory control and renewable harvesting illustrate the theory.
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