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.

arXiv commentarXiv admin note: text overlap with arXiv:1307.8284 by other authors