The sharp iid Berry–Esseen inequality: Analytic reductions and exact certificate proof
Let $X_1, X_2, \dots$ be independent and identically distributed real random variables with $\mathbb{E}[X_1] = 0$, $\mathbb{E}[X_1^2] = 1$, and finite third absolute moment $\rho = \mathbb{E}|X_1|^3 < \infty$. Write $F_n(x) = \mathbb{P}((X_1 + \cdots + X_n)/\sqrt{n} \le x)$ and let $\Phi$ denote the standard normal distribution function. Define \[ \Delta_n = \sup_{x \in \mathbb{R}} |F_n(x) - \Phi(x)|, \qquad C_n = \sup \frac{\sqrt{n}\,\Delta_n}{\rho}, \qquad C^* = \sup_{n \ge 1} C_n, \] and let \[ C_{\mathrm{E}} = \frac{3 + \sqrt{10}}{6\sqrt{2\pi}} \approx 0.4097321837 \] denote Esseen's…
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