Math Conjectures

Optimal i.i.d. Berry–Esseen constant

OPENMajorExact constant problemProposed 1942 · Standard version

Canonical statement

Let Φ\Phi be the standard normal distribution function. For i.i.d. real-valued random variables X1,…,XnX_1,\ldots,X_n, define
CBE=sup⁡n≥1, X1,…,Xn i.i.d.EX1=0, EX12=1, 0<ρ:=E∣X1∣3<∞nρsup⁡x∈R∣P ⁣{X1+⋯+Xnn≤x}−Φ(x)∣. C_{\mathrm{BE}}=\sup_{\substack{n\ge1,\ X_1,\ldots,X_n\ \mathrm{i.i.d.}\\ \mathbf E X_1=0,\ \mathbf E X_1^2=1,\ 0<\rho:=\mathbf E|X_1|^3<\infty}} \frac{\sqrt n}{\rho}\sup_{x\in\mathbb R}\left|\mathbf P\!\left\{\frac{X_1+\cdots+X_n}{\sqrt n}\le x\right\}-\Phi(x)\right|.
Determine CBEC_{\mathrm{BE}} exactly.
View source LaTeX
Let \(\Phi\) be the standard normal distribution function. For i.i.d. real-valued random variables \(X_1,\ldots,X_n\), define
\[
C_{\mathrm{BE}}=\sup_{\substack{n\ge1,\ X_1,\ldots,X_n\ \mathrm{i.i.d.}\\ \mathbf E X_1=0,\ \mathbf E X_1^2=1,\ 0<\rho:=\mathbf E|X_1|^3<\infty}}
\frac{\sqrt n}{\rho}\sup_{x\in\mathbb R}\left|\mathbf P\!\left\{\frac{X_1+\cdots+X_n}{\sqrt n}\le x\right\}-\Phi(x)\right|.
\]
Determine \(C_{\mathrm{BE}}\) exactly.
STAT-001:[Esseen1942Fourier][Shevtsova2013Absolute][Pinelis2017Nonuniform]
StatisticsProbability theory

Notes

The Berry–Esseen theorem quantifies the central limit theorem: for i.i.d. random variables with mean zero, unit variance, and finite third absolute moment ρ\rho, the Kolmogorov distance between the law of the normalized sum and the standard normal distribution is at most Cρ/nC\rho/\sqrt n for an absolute constant CC. The problem asks for the exact value of the optimal constant CBEC_{\mathrm{BE}} in this i.i.d. setting. It goes back to Esseen's 1942 analysis of the error in Liapounoff's limit theorem [Esseen1942Fourier].

The two sides of the question have advanced very unevenly. Esseen's two-point distributions show CBE≥(3+10)/(62π)=0.4097…C_{\mathrm{BE}}\ge(3+\sqrt{10})/(6\sqrt{2\pi})=0.4097\ldots, but whether this bound is sharp is not known. In the other direction, successive refinements of the Fourier-analytic method have brought the general i.i.d. upper bound down to the commonly quoted value 0.46900.4690 [Shevtsova2013Absolute]; related nonuniform versions of the inequality have been sharpened as well [Pinelis2017Nonuniform].

The problem remains open: a full solution requires closing the gap between 0.4097…0.4097\ldots and 0.46900.4690, identifying the exact constant, and describing all extremizing distributions.

References (3)

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