This paper establishes central limit theorems for Polyak-Ruppert averaged Q-learning under asynchronous updates. We prove a non-asymptotic central limit theorem, where the convergence rate in Wasserstein distance explicitly reflects the dependence on the number of iterations, state-action space size, the discount factor, and the quality of exploration. In addition, we derive a functional central limit theorem, showing that the partial-sum process converges weakly to a Brownian motion.

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