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

Let be the group of -adic numbers, and let and be independent random variables with values in and distributions and . Let be topological automorphisms of . Assuming that the linear forms and are independent, we describe possible distributions and depending on the automorphisms . This theorem is an analogue for the group of the well-known Skitovich-Darmois theorem, where a Gaussian distribution on the real line is characterized by the independence of two linear forms.