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

Author

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  • Gennadiy Feldman

    (National Academy of Sciences of Ukraine)

  • Margaryta Myronyuk

    (National Academy of Sciences of Ukraine)

Abstract

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

Suggested Citation

  • Gennadiy Feldman & Margaryta Myronyuk, 2017. "On a Characterization of Idempotent Distributions on Discrete Fields and on the Field of p-Adic Numbers," Journal of Theoretical Probability, Springer, vol. 30(2), pages 608-623, June.
    • Handle: RePEc:spr:jotpro:v:30:y:2017:i:2:d:10.1007_s10959-015-0657-1
    DOI: 10.1007/s10959-015-0657-1
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    References listed on IDEAS

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    1. Gennadiy Feldman, 2015. "On the Skitovich–Darmois Theorem for the Group of $$p$$ p -Adic Numbers," Journal of Theoretical Probability, Springer, vol. 28(2), pages 539-549, June.
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