## Some estimate of the rate of convergence to the normal law.(Ukrainian, English)Zbl 1050.60027

Teor. Jmovirn. Mat. Stat. 68, 112-114 (2003); translation in Theory Probab. Math. Stat. 68, 123-126 (2004).
Let $$\xi_{1},\xi_2,\ldots$$ be a sequence of independent random variables with distribution functions $$F_{i}(x)$$, $$E[\xi_{i}]=0$$, $$D[\xi_{i}]=\sigma_{i}^2$$, $$i\geq1$$. Let us denote by $$\Phi_{n}(x)$$ the distribution function of the random variable $$(\xi_{1}+\ldots+\xi_{n})/B_{n}$$, $$B_{n}=\sigma_{1}^2+\ldots+\sigma_{n}^2$$, and let $$\overline\sigma_{i}=\min(1,\sigma_{i})$$, $$\overline B_{n}^2=\overline\sigma_{1}^2+\ldots+\overline\sigma_{n}^2$$. The author proves that for all $$n\geq1$$: $$\rho_{n}\leq C({1\over\overline B_{n}}\Lambda_{n}^{(1)}+\Lambda_{n}^{(2)})b_{n}^{-3}$$, where $$\rho_{n}=\sup_{x}|\Phi_{n}(x)-\Phi(x)|$$, $$\Phi(x)$$ is the distribution function of the normal law, $$C$$ is a constant,
\begin{aligned} \lambda_{ni}^{(1)} &= \int_{| x|\leq B_{n}}\max(1,| x|^3)| d(F_{i}(x)-\Phi(x/ \sigma_{i}))|,\\ \lambda_{ni}^{(2)} &= \int\! sb{| x|\leq B_{n}}x^2| d(F_{i}(x)-\Phi(x/ \sigma_{i}))|,\\ \Lambda_{n}^{(1)} &={1\over\overline B_{n}^2}\sum_{i=1}^{n}\lambda_{ni}^{(1)},\quad \Lambda_{n}^{(2)}= {1\over\overline B_{n}^2}\sum_{i=1}^{n}\lambda_{ni}^{(2)}, \quad b_{n} = \min(\overline\sigma_1,\ldots,\overline\sigma_{n}). \end{aligned}

### MSC:

 60F10 Large deviations

### Keywords:

estimates; rate of convergence; normal law