Independent linear forms on the group ${\mathrm{\Omega }}_p$

Let ${\mathrm{\Omega }}_p$ be the group of $p$-adic numbers, ${\xi }_1$, ${\xi }_2$, ${\xi }_3$ be independent random variables with values in ${\mathrm{\Omega }}_p$ and distributions ${\mu }_1$, ${\mu }_2$, ${\mu }_3$. Let ${\alpha }_j,{\beta }_j,{\gamma }_j$ be topological automorphisms of ${\mathrm{\Omega }}_p$. We consider linear forms $L_1 = \alpha_1\xi_1 + \alpha_2 \xi_2+\alpha_3 \xi_3$,$L_2=\beta_1\xi_1 + \beta_2 \xi_2+ \beta_3 \xi_3$and$L_3=\gamma_1\xi_1 + \gamma_2 \xi_2+ \gamma_3 \xi_3$. Assuming that the linear forms $L_1$, $L_2$ and $L_3$ are independent, we describe possible distributions ${\mu }_1$, ${\mu }_2$, ${\mu }_3$. 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.