On the Skitovich–Darmois Theorem for the Group of $$p$$ 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.