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on Berry–Esseen constant ($C_{19}$)

The statement

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).

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.