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On a generalisation of the Skitovich–Darmois theorem for several linear forms on Abelian groups

  • Gennadiy FeldmanEmail author
Article
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Abstract

A.M. Kagan introduced a class of distributions \(\mathcal {D}_{m, k}\) in \(\mathbb {R}^m\) and proved that if the joint distribution of m linear forms of n independent random variables belongs to the class \(\mathcal {D}_{m, m-1}\), then the random variables are Gaussian. A.M. Kagan’s theorem implies, in particular, the well-known Skitovich–Darmois theorem, where the Gaussian distribution on the real line is characterized by independence of two linear forms of n independent random variables. In the note we describe a wide class of locally compact Abelian groups where A.M. Kagan’s theorem is valid.

Keywords

Locally compact Abelian group Gaussian distribution Linear forms 

Mathematics Subject Classification

43A25 43A35 60B15 62E10 

Notes

Acknowledgements

The author thanks the reviewer for carefully reading the article and the language corrections.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of UkraineKharkivUkraine

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