A non-uniform estimation of the remainder term in the central limit theorem for scheme of series. (Russian) Zbl 0576.60018

Ignat, Yu. I.

Teor. Veroyatn. Mat. Stat. 31, 40-44 (1984).

Let \(\{X_{nk};k=1,...,k_ n;n=1,2,...\}\) be a triangular array of row- wis independent random variables with the corresponding array \(\{F_{nk}\}\) of distribution functions and assume that \(EX_{nk}\equiv 0\) and \(\sum^{k_ n}_{k=1}EX^ 2_{nk}\equiv 1\). Under the Lindeberg condition the author proves that \[ | P\{\sum^{k_ n}_{k=1}X_{nk}<x\}-\Phi (x)| \frac{C}{1+x^ 2}\frac{1\quad}{| x|}\int^{| x|}_{0}\{\sum^{k_ n}_{k=1}\int_{| y| >t}y^ 2dF_{nk\quad}(y)\}dt, \] where \(\Phi\) is the standard normal distribution function and C is an absolute constant. A result on the \(L_ p\)-distance, \(p>1/2\), is also formulated as a consequence.

Reviewer: S.Csörgö

MSC:

60F05

Central limit and other weak theorems

Keywords:

central limit theorem; non-uniform bound; triangular array; Lindeberg condition