The response-adaptive design driven by randomly reinforced urn model is optimal in the sense that it allocate patients to the best treatment with probability converging to one. This paper illustrates asymptotic properties for multi-color reinforced urn models. Results on the rate of convergence of the number of patients assigned to each treatment are obtained under minimum requirement of conditions and the distributions of the limits are found. Asymptotic distributions of the Wald test statistic for testing mean differences are obtained both under the null hypothesis and alternate hypothesis. The asymptotic behavior for the non-homogenous is also studied.
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