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

Hoang Huu Nhu

Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 2, pp. 308-314

Résumé

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].