Asymptotically optimal Berry-Esseen-type bounds for distributions with an absolutely continuous part

Recursive and closed form upper bounds are o¤ered for the Kolmogorov and the total variation distance between the standard normal distribution and the distribution of a standardized sum of n independent and identically distributed random variables. The approximation error in the CLT obtained from these new bounds vanishes at a rate O(n ); provided that the common distribution of the summands possesses an absolutely continuous part, and shares the same k 1 (k 3) …rst moments with the standard normal distribution. Moreover, for the …rst time, these new uniform Berry-Esseen-type bounds are asymptotically optimal, that is, the ratio of the true distance to the respective bound converges to unity for a large class of distributions of the summands. Thus, apart from the correct rate, the proposed error estimates incorporate an optimal asymptotic constant (factor). Finally, three illustrative examples are presented along with numerical comparisons revealing that the new bounds are sharp enough even to be used in practical statistical applications. Abbreviated Title: Optimal Berry-Esseen-type bounds. Key words and phrases: Central limit theorem, Berry-Esseen theorem, Edgeworth expansions to the CLT, Rate of convergence, Kolmogorov and total variation distance, Zolotarev’s ideal metric. AMS 2000 subject classi…cation: Primary: 60F05 Secondary: 60G50; 60E15; 62E17; 62G20