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Teor. Veroyatnost. i Primenen., 1968, Volume 13, Issue 2, Pages 308–314 (Mi tvp847)  

Short Communications

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

Hoang Huu Nhuab

a Moscow
b Hanoi

Abstract: 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:
$$ |F(x)-\Phi(\frac{x-a}\sigma)|<\varepsilon,\quad-\infty<x<\infty, $$
where
$$ \Phi(x)=\frac1{\sqrt2\pi}\int_{-\infty}^xe^{-u^2/2}du $$
Let $x_1,…,x_n$ 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 $x_j$ ($j=1,…,n$) are $\delta(\varepsilon)$-normal with the parameters $a$ and $\sigma$, where
$$ \delta(\varepsilon)\le\frac C{\sqrt{\log(\frac1\varepsilon)}}, $$
$C$ 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].

Full text: PDF file (343 kB)

English version:
Theory of Probability and its Applications, 1968, 13:2, 299–304

Bibliographic databases:

Received: 17.12.1966

Citation: Hoang Huu Nhu, “On the stability of some theorems of characterization of the normal population”, Teor. Veroyatnost. i Primenen., 13:2 (1968), 308–314; Theory Probab. Appl., 13:2 (1968), 299–304

Citation in format AMSBIB
\Bibitem{Hoa68}
\by Hoang~Huu~Nhu
\paper On the stability of some theorems of characterization of the normal population
\jour Teor. Veroyatnost. i Primenen.
\yr 1968
\vol 13
\issue 2
\pages 308--314
\mathnet{http://mi.mathnet.ru/tvp847}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=234548}
\zmath{https://zbmath.org/?q=an:0167.47404|0165.21803}
\transl
\jour Theory Probab. Appl.
\yr 1968
\vol 13
\issue 2
\pages 299--304
\crossref{https://doi.org/10.1137/1113033}


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