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

Abstract

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