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Let $(X_1,Y_1),...,(X_m,Y_m)$ be $m$ independent identically distributed bivariate vectors and $L_1={\beta }_1{X}_1+...+{\beta }_m{X}_m$, $L_2={\beta }_1{Y}_1+...+{\beta }_m{Y}_m$ are two linear forms with positive coefficients. We study two problems: under what conditions does the equidistribution of $L_1$ and $L_2$ imply the same property for $X_1$ and $Y_1$, and under what conditions does the independence of $L_1$ and $L_2$ entail independence of $X_1$ and $Y_1$? Some analytical sufficient conditions are obtained and it is shown that in general they can not be weakened.

Let () be independent identically distributed bivariate vectors and , are two linear forms with positive coefficients. We study two problems: under what conditions does the equidistribution of and imply the same property for and , and under what conditions does the independence of and entail independence of and ? Some analytical sufficient conditions are obtained...