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We prove optimal convergence rates in the central limit theorem for sums Rᵏ. Assuming a fourth moment, we obtain a Berry-Esseen type bound of O(N⁻¹) for the probability of hitting a ball provided that k≤ 5. The proof still requires a technical assumption related to the independence of coordinate sums.
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Bentkus et al. (1996) studied this question.
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