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Let $X_1,\ldots,X_n$ be independent real random variables with $\mathbb EX_i=0$, finite $\rho_i=\mathbb E|X_i|^3$, and total variance $V=\sum_i\mathbb EX_i^2>0$. If $\Phi$ is the standard normal distribution function, then for an absolute constant $C$,

$$\sup_{t\in\mathbb R} \left|\Pr\left(V^{-1/2}\sum_iX_i\le t\right)-\Phi(t)\right| \le C\frac{\sum_i\rho_i}{V^{3/2}}.$$

This is an external statement, not a proof reconstructed here. The paper cites Berry (1941), Esseen (1942), and Shevtsova (2010); it does not need a particular numerical value of $C$. The condition $V>0$ is stated explicitly because the normalized variable is otherwise undefined.

The local moment calculation verifies every hypothesis, including uniformity after removing one coordinate, in the application.

Source: published PDF, p. 4, Lemma 3; earlier v1 p. 3, Lemma 4.

Bears on. Problem 297.