Bounds on the constant in the mean central limit theorem
Larry Goldstein
Source: Ann. Probab. Volume 38, Number 4
(2010), 1672-1689.
Abstract
Let X1, …, Xn be independent with zero means, finite variances σ12, …, σn2 and finite absolute third moments. Let Fn be the distribution function of (X1 + ⋯ + Xn)/σ, where σ2 = ∑i=1nσi2, and Φ that of the standard normal. The L1-distance between Fn and Φ then satisfies
In particular, when X1, …, Xn are identically distributed with variance σ2, we have
for all n ∈ ℕ,
corresponding to an L1-Berry–Esseen constant of 1.
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