Multi-armed bandit (MAB) processes constitute a foundational subclass of reinforcement learning problems and represent a central topic in statistical decision theory, but are limited to simultaneous adaptive allocation and sequential test, because of the absence of asymptotic theory under non-i.i.d sequence and sublinear information. To address this open challenge, we propose Urn Bandit (UNB) process to integrate the reinforcement mechanism of urn probabilistic models with MAB principles, ensuring almost sure convergence of resource allocation to optimal arms. We establish the joint functional central limit theorem (FCLT) for consistent estimators of expected rewards under non-i.i.d., non-sub-Gaussian and sublinear reward samples with pairwise correlations across arms. To overcome the limitations of existing methods that focus mainly on cumulative regret, we establish the asymptotic theory along with adaptive allocation that serves powerful sequential test, such as arms comparison, A/B testing, and policy valuation. Simulation studies and real data analysis demonstrate that UNB maintains statistical test performance of equal randomization (ER) design but obtain more average rewards like classical MAB processes.

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