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An analogue of the Bernstein theorem for the cylinder

Summary.

In 1941 S. N. Bernstein proved the following characterization theorem. Let ξ1 and ξ2 be independent random variables. If ξ1 + ξ2 and ξ1 - ξ2 are independent, then ξ1 and ξ2 are Gaussian with equal dispersion. This paper is devoted to a group analogue of the Bernstein theorem. Let X be a locally compact separable abelian group and ξ1, ξ2 be independent random X-valued variables with distributions μ1, μ2. Denote by \( \user1{\mathbb{T}} \) and by ∑ a the group of rotations of the circle and the a-adic solenoid, respectively. In this paper we solve the following problem: In the case where either \( X = \user1{\mathbb{R}} \times \user1{\mathbb{T}} \) or X = ∑ a we describe all possible distributions μ1 and μ2 under the assumption that ξ1 + ξ2 and ξ1 - ξ2 are independent.

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Correspondence to Margaryta Myronyuk.

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Manuscript received: February 2, 2004 and, in final form, November 29, 2004.

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Myronyuk, M. An analogue of the Bernstein theorem for the cylinder. Aequ. math. 71, 54–69 (2006). https://doi.org/10.1007/s00010-005-2773-y

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  • DOI: https://doi.org/10.1007/s00010-005-2773-y

Mathematics Subject Classification (2000).

  • 39B99
  • 60B15
  • 62E10

Keywords.

  • Functional equation
  • Bernstein’s characterization theorem
  • Gaussian distribution
  • abelian group