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

Gennadiy Feldman () and Margaryta Myronyuk ()
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Gennadiy Feldman: National Academy of Sciences of Ukraine
Margaryta Myronyuk: National Academy of Sciences of Ukraine

Journal of Theoretical Probability, 2017, vol. 30, issue 2, 608-623

Abstract: 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.

Keywords: Characterization theorem; Idempotent distribution; Discrete field; The field of p-adic numbers; 60B15; 62E10; 43A05 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10959-015-0657-1

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