On a refinemet of the central limit theorem and its global version

Yu. P. Studnev   ; Yu. I. Ignat

Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 3, pp. 562-567

Résumé

Let $\{\xi_k\}$ be a sequence of independent random variables with zero means and $\{\sigma_k^{(2)}\}$ be the sequence of their variances. Denote \begin{gather*} s_n=\frac{\xi_1+\dots+\xi_n}{B_n},\quad B_n=\sum_{k=1}^n\sigma_k^2,\quad\Phi_n(x)=\mathbf P(s_n<x) \\ L_n(x)=\frac1{B_n^2}\sum_{k=1}^n\int_{|z|>x}z^2\,dF_k(z),\quad\Phi(x)=\frac1{\sqrt{2\pi}}\int_{-\infty}^xe^{-x^2/2}\,dt. \end{gather*} The main result of the paper is the following. Theorem. {\it Under Lindeberg's condition in the central limit theorem the inequality $$ |\Phi_n(x)-\Phi(x)|<C\min\biggl\{\frac1{B_n}\int_0^{B_n}L_n(x)\,dx,\quad\frac{\frac1{|x|B_n}\int_0^{|x|B_n}L_n(x)\,dx}{1+x^2}\biggr\}, $$ holds true where $C$ is an absolute constant}.