2017
Том 69
№ 9

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Skitovich-Darmois theorem for finite Abelian groups

Mazur I. P.

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Abstract

Let $X$ be a finite Abelian group, let $\xi_i,\; i = 1, 2, . . . , n,\; n ≥ 2$, be independent random variables with values in $X$ and distributions $\mu_i$, and let $\alpha_{ij},\; i, j = 1, 2, . . . , n$, be automorphisms of $X$. We prove that the independence of n linear forms $L_j = \sum_{i=1}^{n} \alpha_{ij} \xi_i$ implies that all $\mu_i$ are shifts of the Haar distributions on some subgroups of the group $X$. This theorem is an analog of the Skitovich – Darmois theorem for finite Abelian groups.

English version (Springer): Ukrainian Mathematical Journal 63 (2011), no. 11, pp 1719-1732.

Citation Example: Mazur I. P. Skitovich-Darmois theorem for finite Abelian groups // Ukr. Mat. Zh. - 2011. - 63, № 11. - pp. 1512-1523.

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