Asymptotic Refinement of the Berry-Esseen Constant

For sum of independent and identically distributed (i.i.d.) random variables {Xi}i=1, the Berry-Esseen theorem states that sup y∈< ∣∣∣∣Pr { 1 sn (X1 + X2 + · · ·+ Xn) ≤ y } − Φ(y) ∣∣∣∣ ≤ C ρ σ3 √ n , where σ and ρ are respectively the variance and the absolute third moment of the parent distribution, Φ(·) is the unit normal cumulative distribution function, and C is an absolute constant. In this work, we re-examined the above inequality by following similar procedure as in [3, Sec.XVI.5, Thm. 1]. Instead of targeting an absolute constant C, we sought for a sample-size-dependent coefficient Cn such that ∣∣∣∣Pr [ 1 σ √ n (X1 + · · ·+ Xn) ≤ y ] − Φ(y) ∣∣∣∣ ≤ Cn ρ σ3 √ n (1) hold for every sample size n. Based on the new standpoint, we found that Cn can be made smaller than Shiganov’s constant 0.7655 when n ≥ 65, and can be decreased by further increasing sample size n. As n approaches infinity, the Berry-Esseen constant can be asymptotically improved down to 0.7164.