The Berry–Esseen constant

Description of constant

Let $X_{1},X_{2},\dots$ be i.i.d. real random variables with $\mathbb E X_{1} = 0$, $\mathrm{Var}(X_{1})=1$, and finite third absolute moment

\[\beta_{3} := \mathbb E|X_{1}|^3 < \infty.\]

Let

\[S_n := \frac{X_{1}+\cdots+X_{n}}{\sqrt n},\qquad F_n(x):=\mathbb P(S_n\le x),\]

and let $\Phi$ denote the standard normal distribution function.

We define $C_{19}$ to be the smallest constant such that the classical Berry–Esseen inequality

\[\Delta_n := \sup_{x\in\mathbb R}\bigl|F_n(x)-\Phi(x)\bigr| \ \le\ C_{19}\,\frac{\beta_{3}}{\sqrt n} \qquad\text{for all } n\ge 1\]

holds for all such distributions of $X_{1}$.

This constant is also called the absolute constant $C_{0}$ in the Berry–Esseen inequality (for i.i.d. summands).

Known upper bounds

Bound	Reference	Comments
$0.82$	[Z1967]	Zolotarev-type smoothing inequalities; [Z1967] also gives $0.9051$ for the general (non-i.i.d.) case.
$0.7975$	[vB1972]	Fourier-analytic refinement.
$0.7655$	[Shi1986]
$0.7056$	[She2006]
$0.5129$	[KS2009]	From the structural bound $\Delta_{n} \le 0.34445(\beta_{3}+0.489)/\sqrt n$.
$0.4785$	[Tyu2009]
$0.4748$	[She2011]
$0.4690$	[She2013]

Known lower bounds

Bound	Reference	Comments
$c_{E} := \dfrac{\sqrt{10}+3}{6\sqrt{2\pi}} \approx 0.4097321837$	[E1956]	Esseen’s lower bound; achieved asymptotically by (centered/normalized) Bernoulli sums.

Additional comments

Zolotarev’s conjecture: It is conjectured that $C_{19}=c_{E}$, i.e. the Esseen lower bound is sharp. This is sometimes attributed to Zolotarev (1967). See e.g. [vB1972], [KS2009], [Tyu2009] for discussion.
How the bound $0.4690$ is obtained: Shevtsova [She2013] proves the structural estimate
\[\Delta_n \le \frac{1}{\sqrt n}\min\bigl\{0.4690,\beta_{3},\ 0.3322(\beta_{3}+0.429),\ 0.3031(\beta_{3}+0.646)\bigr\},\]
which in particular implies $C_{19}\le 0.4690$.
Binomial/Bernoulli case: In the special case of i.i.d. Bernoulli summands (equivalently, binomial distributions after normalization), the optimal constant is known to equal $c_{E}$; see [Sch2016] and references therein.

References

[E1956] Esseen, Carl-Gustav. A moment inequality with an application to the central limit theorem. Skand. Aktuarietidskr. 39 (1956), 160–170.
[KS2009] Korolev, V. Yu.; Shevtsova, I. G. On the upper bound for the absolute constant in the Berry–Esseen inequality. Teor. Veroyatn. Primen. 54 (2009), no. 4, 671–695 (English transl.: Theory Probab. Appl. 54 (2010), no. 4, 638–658).
[Sch2016] Schulz, Jona. The optimal Berry–Esseen constant in the binomial case. PhD thesis, Universität Trier (2016).
[She2006] Shevtsova, I. G. A refinement of the upper estimate of the absolute constant in the Berry–Esseen inequality. Teor. Veroyatn. Primen. 51 (2006), no. 3, 622–626 (English transl.: Theory Probab. Appl. 51 (2007), 549–553).
[She2011] Shevtsova, Irina. On the absolute constants in the Berry–Esseen type inequalities for identically distributed summands. arXiv:1111.6554 (2011).
[She2013] Shevtsova, I. G. On the absolute constants in the Berry–Esseen inequality and its structural and nonuniform improvements. Inform. Primen. 7 (2013), no. 1, 124–125.
[Shi1986] Shiganov, I. S. Refinement of the upper bound of the constant in the central limit theorem. J. Soviet Math. 35 (1986), 2545–2550.
[Tyu2009] Tyurin, I. S. New estimates of the convergence rate in the Lyapunov theorem. arXiv:0912.0726 (2009).
[vB1972] van Beek, Paul. An application of Fourier methods to the problem of sharpening the Berry–Esseen inequality. Z. Wahrscheinlichkeitstheorie verw. Geb. 23 (1972), 187–196.
[Z1967] Zolotarev, V. M. A sharpening of the inequality of Berry–Esseen. Z. Wahrscheinlichkeitstheorie verw. Geb. 8 (1967), 332–342.

Acknowledgements

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