On the stability of certain characteristic properties of the normal distribution

Yu. R. Gabovich

Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 2, pp. 374-382

Résumé

Consider a set of the random variables (r.v.) $X_1,\dots,X_n$ with the joint distribution function (d.f.) $F$, marginal d.f. $F_1,\dots,F_n$ and the Lévy metric in $R^n$. We denote by $N$ the set of $n$-dimensional normal d.f. D e f i n i t i o n 1. The r.v. $X_1,\dots,X_n$ are $(L,\varepsilon)$-independent if $L(F,F_1,\dots,F_n)\le\varepsilon$. D e f i n i t i o n 2. The r.v. $X_1,\dots,X_n$ are $(L,\delta)$-normal if $\inf\limits_NL(F,\Phi)\le\delta$. Let $\gamma(\varepsilon)=\sup\limits_{\mathfrak M_\varepsilon}\inf\limits_NL(F,\Phi)$, where $\mathfrak M_\varepsilon$ consists of the d.f. $F$ of r.v. $X_1,\dots,X_n$ such that $\mathbf P\{|X_j|>d\}<q$, $j=1,\dots,n$, and that $X_1,\dots,X_n$ are $(L,\varepsilon)$-independent, and $L_1=X_1+\dots+X_n$, $L_2=b_1X_1+\dots+b_nX_n$, $b_j\ne0$, $j=1,\dots,n$, $\sum_1^nb_j=0$ are $(L,\varepsilon)$-independent.