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On the Skitovich–Darmois theorem for some locally compact Abelian groups

Abstract

Let X be a locally compact Abelian group, \(\alpha _{j}, \beta _j\) be topological automorphisms of X. Let \(\xi _1, \xi _2\) be independent random variables with values in X and distributions \(\mu _j\) with non-vanishing characteristic functions. It is known that if X contains no subgroup topologically isomorphic to the circle group \(\mathbb {T}\), then the independence of the linear forms \(L_1=\alpha _1\xi _1+\alpha _2\xi _2\) and \(L_2=\beta _1\xi _1+\beta _2\xi _2\) implies that \(\mu _j\) are Gaussian distributions. We prove that if X contains no subgroup topologically isomorphic to \(\mathbb {T}^2\), then the independence of \(L_1\) and \(L_2\) implies that \(\mu _j\) are either Gaussian distributions or convolutions of Gaussian distributions and signed measures supported in a subgroup of X generated by an element of order 2. The proof is based on solving the Skitovich–Darmois functional equation on some locally compact Abelian groups.

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Correspondence to Gennadiy Feldman.

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Feldman, G., Myronyuk, M. On the Skitovich–Darmois theorem for some locally compact Abelian groups. Aequat. Math. 92, 1129–1147 (2018). https://doi.org/10.1007/s00010-018-0580-5

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  • DOI: https://doi.org/10.1007/s00010-018-0580-5

Keywords

  • Locally compact Abelian group
  • Gaussian distribution
  • Independent linear forms
  • Skitovich–Darmois functional equation

Mathematics Subject Classification

  • 62E10
  • 39B52
  • 60B15