On nonuniform estimates of the rate of convergence in the central limit theorem

It is shown that in the nonuniform analog of the Berry–Esseen inequality $(1+|x|^3)|F_n(xB_n)-\Phi(x)|\le \left(C/{B_n^3}\right)\sum\limits_{k=1}^n\beta_k$, $n\ge1$, $x\in\mathbb R$, where $F_n(x)$ is the distribution function of the sum of $n$ independent random variables $X_1, \dots ,X_n$ with $E X_k=0$, $E X_k^2=\sigma_k^2$; $\beta_k=E |X_k|^3<\infty$, $k=1,\dots,n$; $B_n^2=\sigma_1^2+\dotsb+\sigma_n^2$; $\Phi(x)$ is the standard normal distribution function, the absolute constant $C$ satisfies the inequality $C\le 22.2417$.