On the Skitovich–Darmois Theorem for the Group of $$p$$ p -Adic Numbers

Gennadiy Feldman ()
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Gennadiy Feldman: National Academy of Sciences of Ukraine

Journal of Theoretical Probability, 2015, vol. 28, issue 2, 539-549

Abstract: Abstract Let $$\Omega _p$$ Ω p be the group of $$p$$ p -adic numbers, and let $$\xi _1$$ ξ 1 and $$\xi _2$$ ξ 2 be independent random variables with values in $$\Omega _p$$ Ω p and distributions $$\mu _1$$ μ 1 and $$\mu _2$$ μ 2 . Let $$\alpha _j, \beta _j$$ α j , β j be topological automorphisms of $$\Omega _p$$ Ω p . Assuming that the linear forms $$L_1=\alpha _1\xi _1 + \alpha _2\xi _2$$ L 1 = α 1 ξ 1 + α 2 ξ 2 and $$L_2=\beta _1\xi _1 + \beta _2\xi _2$$ L 2 = β 1 ξ 1 + β 2 ξ 2 are independent, we describe possible distributions $$\mu _1$$ μ 1 and $$\mu _2$$ μ 2 depending on the automorphisms $$\alpha _j, \beta _j$$ α j , β j . This theorem is an analogue for the group $$\Omega _p$$ Ω p of the well-known Skitovich–Darmois theorem, where a Gaussian distribution on the real line is characterized by the independence of two linear forms.

Keywords: Group of $$p$$ p -adic numbers; Characterization theorem; 60B15; 62E10; 43A35 (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s10959-013-0525-9

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