A Group Analogue of the Ghurye--Olkin--Ibragimov Theorem
Abstract
According to the classical Skitovich--Darmois theorem, the Gaussian distribution on the real line is characterized by the independence of two linear forms of a finite number of independent random variables ξj. This theorem has been extended in various directions. In particular, S.G. Ghurye and I. Olkin established an analogous result for the case where ξj are n-dimensional independent random vectors and the coefficients of the linear forms are invertible n× n matrices. Subsequently, A.A. Zinger and later I.A. Ibragimov investigated linear forms of an infinite sequence of n-dimensional independent random vectors. In the present paper, we investigate, for the first time, linear forms of a sequence of independent random variables taking values in a second-countable locally compact Abelian group X under either of the following conditions: X contains no nontrivial compact subgroups, or X contains no subgroups topologically isomorphic to the circle group and has a finite topological automorphism group. Furthermore, we investigate the case where the independent random variables take values in an a-adic solenoid Σa. In all these settings, the coefficients of the linear forms are topological automorphisms of the corresponding group.
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BibTeX
@misc{feldman2026group,
title = {{A Group Analogue of the Ghurye--Olkin--Ibragimov Theorem}},
author = {Gennadiy Feldman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19346},
eprint = {2609.19346},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19346},
}
RIS
TY - UNPB
TI - A Group Analogue of the Ghurye--Olkin--Ibragimov Theorem
AU - Gennadiy Feldman
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.19346
UR - https://arxiv.org/abs/2609.19346
AB - According to the classical Skitovich--Darmois theorem, the Gaussian distribution on the real line is characterized by the independence of two linear forms of a finite number of independent random variables ξ_j. This theorem has been extended in various directions. In particular, S.G. Ghurye and I. Olkin established an analogous result for the case where ξ_j are n-dimensional independent random vectors and the coefficients of the linear forms are invertible n× n matrices. Subsequently, A.A. Zinger and later I.A. Ibragimov investigated linear forms of an infinite sequence of n-dimensional independent random vectors. In the present paper, we investigate, for the first time, linear forms of a sequence of independent random variables taking values in a second-countable locally compact Abelian group X under either of the following conditions: X contains no nontrivial compact subgroups, or X contains no subgroups topologically isomorphic to the circle group and has a finite topological automorphism group. Furthermore, we investigate the case where the independent random variables take values in an a-adic solenoid Σ_a. In all these settings, the coefficients of the linear forms are topological automorphisms of the corresponding group.
ER -