Geodesic
On the Skitovich--Darmois theorem for discrete abelian groups
Teoriâ veroâtnostej i ee primeneniâ, Tome 49 (2004) no. 3, pp. 596-601

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The following theorem is proved: Let $X$ be a discrete countable Abelian group, let $\xi_1,\xi_2$ be independent random variables with values in the group $X$ and with distributions $\mu_1,\mu_2$, and let $\alpha_j,\beta_j$, $j=1, 2$, be automorphisms of the group $X$. Then the independence of the linear statistics $L_1=\alpha_1\xi_1 + \alpha_2\xi_2$ and $L_2=\beta_1\xi_1 + \beta_2\xi_2$ implies that $\mu_1$ and $\mu_2$ are idempotent distributions.
Keywords: independent linear statistics, discrete Abelian group, Skitovich–Darmois theorem. 
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     author = {G. M. Feldman and P. Graczyk},
     title = {On the {Skitovich--Darmois} theorem for discrete abelian groups},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {596--601},
     publisher = {mathdoc},
     volume = {49},
     number = {3},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2004_49_3_a9/}
}
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G. M. Feldman; P. Graczyk. On the Skitovich--Darmois theorem for discrete abelian groups. Teoriâ veroâtnostej i ee primeneniâ, Tome 49 (2004) no. 3, pp. 596-601. http://geodesic.mathdoc.fr/item/TVP_2004_49_3_a9/