The Skitovich--Darmois theorem for discrete periodic Abelian groups
Teoriâ veroâtnostej i ee primeneniâ, Tome 42 (1997) no. 4, pp. 747-756
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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.
@article{TVP_1997_42_4_a5,
author = {G. M. Feldman},
title = {The {Skitovich--Darmois} theorem for discrete periodic {Abelian} groups},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {747--756},
publisher = {mathdoc},
volume = {42},
number = {4},
year = {1997},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1997_42_4_a5/}
}
G. M. Feldman. The Skitovich--Darmois theorem for discrete periodic Abelian groups. Teoriâ veroâtnostej i ee primeneniâ, Tome 42 (1997) no. 4, pp. 747-756. http://geodesic.mathdoc.fr/item/TVP_1997_42_4_a5/