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

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Let Ωₚ be the group of p-adic numbers, ξ₁ and ξ₂ be independent random variables with values in Ωₚ and distributions μ₁ and μ₂. Let αⱼ, βⱼ be topological automorphisms of Ωₚ. 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 αⱼ, βⱼ. This theorem is an analogue for the group Ωₚ of the well-known Skitovich--Darmois theorem, where a Gaussian distribution on the real line is characterized by the independence of two linear forms.