Bounds on the constant in the mean central limit theorem

By Larry Goldstein. 2009

56 citations recorded by the source corpus.

Let X 1 , ..., X n be independent with zero means, finite variances σ 2 1 , ..., σ 2 n and finite absolute third moments. Let F n be the distribution function of (X 1 + ... + X n )/σ, where σ 2 = Σ n i=1 σ 2 i , and Φ that of the standard normal. The L 1 -distance between F n and Φ then satisfies ∥F n ― Φ∥ 1 ≤ 1 σ 3 E|X i | 3 . In particular, when X 1 , ..., X n are identically distributed with variance σ 2 , we have ∥F n ― Φ∥ 1 ≤ E|X 1 | 3 / σ√ 3 n for all n ∈ ℕ, corresponding to an L 1 -Berry―Esseen constant of 1.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a37fb968a0809cbcee752f2d6637bd3ed9ef317e