Short Communications
On a refinemet of the central limit theorem and its global version
Yu. P. Studnev,
Yu. I. Ignat Uzhgorod
Abstract:
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}.
Received: 10.09.1966
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