This article is cited in 16 papers

The Skitovich–Darmois theorem for discrete periodic Abelian groups

G. M. Feldman

B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine

Abstract: This paper gives a complete description of the periodic Abelian groups $X$ such that the independence of the linear statistics of independent random variables taking values in group $X$ (coefficients of the linear statistics are the automorphisms of group) implies that distributions of the random variables are shifts of the Haar distributions on compact subgroups of group $X$.
For the class of groups under consideration the given theorem is a group analogue of the well-known Skitovich–Darmois theorem on a characterization of the Gaussian distribution by the independence of linear statistics.

Keywords: locally compact Abelian group, Gaussian distribution, idempotent distribution, independent linear statistics.

Received: 24.04.1997

DOI: 10.4213/tvp2183

 English version: Theory of Probability and its Applications, 1997, 42:4, 611–617

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