Specification of some estimates of deviation from the normal law for a sum of independent random variables. (Russian) Zbl 0646.60022

Ignat, Yu. I.; Padits, L.

Teor. Veroyatn. Mat. Stat., Kiev 37, 57-66 (1987).

Let \(\{\xi_{nk}\), \(k=1\), 2,..., \(k_ n\), \(n\in N\}\) be a sequence of series of real, independent random variables, \(S_ n=\sum^{k_ n}_{k=1}\xi_{nk}\), \(E\xi_{nk}=0\). The accuracy of uniform deviation of the distribution function of \(S_ n\) from the normal distribution is investigated. The estimate proved is expressed in terms of moments \[ \gamma_{nk}=\int_{A_{nk}}| u|^3 \,dF_{nk}(u) \] where \(A_{nk}\) is an arbitrary Borel set in \(\mathbb R^1\), and \(F_{nk}\) is the distribution function of \(\xi_{nk}\).

Reviewer: Aleksandras Plikusas

MSC:

60F05	Central limit and other weak theorems
60G50	Sums of independent random variables; random walks

Keywords:

Berry-Esseen estimates; central limit theorem