On the stability of some theorems of characterization of the normal population

In this paper we introduce two definitions:
Definition 1. Two random variables $\xi$ and $\eta$ are said to be $\varepsilon$-independent, if
$$ |\mathbf P\{\xi<x,\ \eta<y\}-\mathbf P\{\xi<x\}\mathbf P\{\eta<y\}|<\varepsilon $$
for all $x$ and $y$, where $\varepsilon$ ($0<\varepsilon<1$) is a given number.
Definition 2. A random variable $\xi$ is said to be $\varepsilon$-normal with the parameters $a,\sigma$ if its distribution function $F(x)$ satisfies the following condition:
$$ \biggl|F(x)-\Phi\biggl(\frac{x-a}\sigma\biggr)\biggr|<\varepsilon,\quad-\infty<x<\infty, $$
where
$$ \Phi(x)=\frac1{\sqrt2\pi}\int_{-\infty}^xe^{-u^2/2}du $$
Let $x_1,\dots,x_n$ be independent sample of size $n$ from a population with a distribution function $F(x)$ and
$$ \mathbf MX_j=a,\quad\mathbf DX_j=\sigma^2,\quad\beta_\delta=\mathbf M|X_j|^{2(1+\delta)},\quad0<\delta\le1. $$

Theorem.\textit{If $\overline x=\frac1n\sum_{j=1}^nx_j$ and $s^2=\frac1n\sum_{j=1}^n(x_i-\overline x)^2$ are $\varepsilon$-independent, then $x_j$ ($j=1,\dots,n$) are $\delta(\varepsilon)$-normal with the parameters $a$ and $\sigma$, where
$$ \delta(\varepsilon)\le\frac C{\sqrt{\log\bigl(\frac1\varepsilon\bigr)}}, $$
$C$ being a constant depending on $\sigma$, $n$, $\delta$ and $\beta_\sigma$.}
A similar result is obtained for the stability of the theorem of S. N. Bernstein [2].