On the Skitovich--Darmois theorem for the group of p-adic numbers

2013/07/31 by Feldman, Gennadiy
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1307.8284

Abstract

Let Ω_p be the group of p-adic numbers, ξ₁ and ξ₂ be independent random variables with values in Ω_p and distributions μ₁ and μ₂. Let α_j, β_j be topological automorphisms of Ω_p. Assuming that the linear forms L₁=α₁ξ₁ + α₂ξ₂ and L₂=β₁ξ₁ + β₂ξ₂ are independent, we describe possible distributions μ₁ and μ₂ depending on the automorphisms α_j, β_j. This theorem is an analogue for the group 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.

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