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On a Characterization of Idempotent Distributions on Discrete Fields and on the Field of p-Adic Numbers

Abstract

We prove the following theorem. Let X be a discrete field, and \(\xi \) and \(\eta \) be independent identically distributed random variables with values in X and distribution \(\mu \). The random variables \(S=\xi +\eta \) and \(D=(\xi -\eta )^2\) are independent if and only if \(\mu \) is an idempotent distribution. A similar result is also proved in the case when \(\xi \) and \(\eta \) are independent identically distributed random variables with values in the field of p-adic numbers \({\mathbf {Q}}_p\), where \(p>2\), assuming that the distribution \(\mu \) has a continuous density.

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Acknowledgments

We would like to thank referees for a very careful reading of the article and for the useful comments and remarks.

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Correspondence to Gennadiy Feldman.

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Feldman, G., Myronyuk, M. On a Characterization of Idempotent Distributions on Discrete Fields and on the Field of p-Adic Numbers. J Theor Probab 30, 608–623 (2017). https://doi.org/10.1007/s10959-015-0657-1

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  • DOI: https://doi.org/10.1007/s10959-015-0657-1

Keywords

  • Characterization theorem
  • Idempotent distribution
  • Discrete field
  • The field of p-adic numbers

Mathematics Subject Classification (2010)

  • 60B15
  • 62E10
  • 43A05