2019
Том 71
№ 9

All Issues

Mazur I. P.

Articles: 2
Article (Russian)

Skitovich–Darmois Theorem for Discrete and Compact Totally Disconnected Abelian Groups

Mazur I. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2013. - 65, № 7. - pp. 946–960

The classic Skitovich–Darmois theorem states that the Gaussian distribution on the real line can be characterized by the independence of two linear forms of n independent random variables. We generalize the Skitovich–Darmois theorem to discrete Abelian groups, compact totally disconnected Abelian groups, and some other classes of locally compact Abelian groups. Unlike the previous investigations, we consider n linear forms of n independent random variables.

Article (Russian)

Skitovich-Darmois theorem for finite Abelian groups

Mazur I. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 11. - pp. 1512-1523

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.