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

Polyak, I. J.

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,
\[ \begin{aligned} \lambda_{ni}^{(1)} &= \int_{| x|\leq B_{n}}\max(1,| x|^3)| d(F_{i}(x)-\Phi(x/ \sigma_{i}))|,\\ \lambda_{ni}^{(2)} &= \int\! sb{| x|\leq B_{n}}x^2| d(F_{i}(x)-\Phi(x/ \sigma_{i}))|,\\ \Lambda_{n}^{(1)} &={1\over\overline B_{n}^2}\sum_{i=1}^{n}\lambda_{ni}^{(1)},\quad \Lambda_{n}^{(2)}= {1\over\overline B_{n}^2}\sum_{i=1}^{n}\lambda_{ni}^{(2)}, \quad b_{n} = \min(\overline\sigma_1,\ldots,\overline\sigma_{n}). \end{aligned} \]

Reviewer: A. D. Borisenko (Kyïv)

MSC:

60F10

Large deviations

Keywords:

estimates; rate of convergence; normal law