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

Let ${\mathrm{\Omega }}_p$ be the group of $p$-adic numbers, ${\xi }_1$ and ${\xi }_2$ be independent random variables with values in ${\mathrm{\Omega }}_p$ and distributions ${\mu }_1$ and ${\mu }_2$. Let ${\alpha }_j,{\beta }_j$ be topological automorphisms of ${\mathrm{\Omega }}_p$. 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 $\alpha_j, \beta_j$\mathrm{.}Thistheoremisananalogueforthegroup$\Omega_p$ of the well-known Skitovich–Darmois theorem, where a Gaussian distribution on the real line is characterized by the independence of two linear forms.