Independent linear forms on the group $Ω_p$

Abstract

Let $Ω_p$ be the group of $p$-adic numbers, $ ξ_1$, $ξ_2$, $ξ_3$ be independent random variables with values in $Ω_p$ and distributions $μ_1$, $μ_2$, $μ_3$. Let $α_j, β_j, γ_j$ be topological automorphisms of $Ω_p$. We consider linear forms $L_1 = α_1ξ_1 + α_2 ξ_2+α_3 ξ_3$, $L_2=β_1ξ_1 + β_2 ξ_2+ β_3 ξ_3$ and $L_3=γ_1ξ_1 + γ_2 ξ_2+ γ_3 ξ_3$. Assuming that the linear forms $L_1$, $L_2$ and $L_3$ are independent, we describe possible distributions $μ_1$, $μ_2$, $μ_3$. This theorem is an analogue of the well-known Skitovich-Darmois theorem, where a Gaussian distribution on the real line is characterized by the independence of two linear forms.