Bounds on the Constant in the Mean Central Limit Theorem
نویسنده
چکیده
Bounds in the mean central limit theorem, where the L1 distance is used to measure the discrepancy of the distribution Fn of a standardized sum of i.i.d. random variables with distributionG from the normal, is of some interest when the normal approximation is to be applied over some wide, and perhaps unspecified range of values. Esseen (1958) showed that the limiting value lim n→∞ n||Fn − Φ||1 = A(G) exists and Zolotarav (1964) provided an explicit representation of A(G) which allowed for the computation of an asymptotic L1 Berry Esseen constant of 1/2. When Fn is the standardized distribution of a sum of independent random variables X1, . . . , Xn with distributions G1, . . . , Gn in F , the collection of non-degenerate mean zero distributions with finite absolute third moments, we show that for all finite n ∈ N, with σ2 = Var(X1 + · · ·+Xn) and G∗ the G-zero biased distribution, ||Fn − Φ||1 ≤ 1 σ3 n ∑ i=1 B(Gi)E|Xi| where B(Gi) = 2EX2 i ||G∗ −G||1 E|Xi| and calculate the supremum of the functional B(G) as sup G∈F B(G) = 1, thus providing the non-asymptotic bound on the mean central limit theorem Berry Esseen constant of 1. A lower bound of 2 √ π(2Φ(1)− 1)− ( √ π + √ 2) + 2e−1/2 √ 2 √ π = 0.535377 . . . on the smallest possible constant is also demonstrated.
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