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

Abstract

Let $Ω_p$ be the group of $p$-adic numbers, $ ξ_1$ and $ξ_2$ be independent random variables with values in $Ω_p$ and distributions $μ_1$ and $μ_2$. Let $α_j, β_j$ be topological automorphisms of $Ω_p$. Assuming that the linear forms $L_1=α_1ξ_1 + α_2ξ_2$ and $L_2=β_1ξ_1 + β_2ξ_2$ are independent, we describe possible distributions $μ_1$ and $μ_2$ 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.