This article is cited in 14 papers

Short Communications

On the Skitovich–Darmois theorem for discrete abelian groups

G. M. Feldman^a, P. Graczyk<sup>b

a</sup> B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine
^b Département de Mathématiques, Université d'Angers

Abstract: 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.

Received: 11.06.2002

DOI: 10.4213/tvp210

 English version: Theory of Probability and its Applications, 2005, 49:3, 527–531

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