An improvement of convergence rate estimates in the Lyapunov theorem (Q542314)

Let \(\;X_{1},X_{2},...,X_{n}\) be independent random variables with \(EX_{i}=0,\) \(DX_{i}=\sigma_{i}^{2}>0,\) \(E\left| X_{i}\right| ^{3}=\beta_{i} <\infty,\) \(i=1,2,\dots,n,\) \(\sum_{i=1}^{n}\sigma_{i}^{2}=1\). The Berry-Esseen inequality gives an estimate for the rate of convergence of the distribution functions \(F_{n}\) of the normalized sums \(S_{n}=X_{1}+X_{2}+\cdot\cdot\cdot+X_{n}\) to the standard normal distribution function \(\Phi(x),\) which has the form \[ \sup_{x\in\mathbb{R}}\left| F_{n}(x)-\Phi(x)\right| \leq C_{0} \sum_{i=1}^{n}\beta_{i}, \] where \(C_{0}\) is an absolute constant. In this paper, the author shows that \[ C_{0}\leq0.56. \]