Independent linear forms on the group $\Omega_p$

Abstract

Let $\Omega_p$ be the group of $p$-adic numbers, $ \xi_1$, $\xi_2$, $\xi_3$ be independent random variables with values in $\Omega_p$ and distributions $\mu_1$, $\mu_2$, $\mu_3$. Let $\alpha_j, \beta_j, \gamma_j$ be topological automorphisms of $\Omega_p$. 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.