On a characterization theorem for the group of p-adic numbers

Abstract

It is well known Heyde's characterization of the Gaussian distribution on the real line: Let ${\xi }_1,{\xi }_2,\dots ,{\xi }_n$, $n\ge 2,$ be independent random variables, let ${\alpha }_j,{\beta }_j$ be nonzero constants such that ${\beta }_i{\alpha }_i^{-1}+{\beta }_j{\alpha }_j^{-1}\ne 0$ for all $i\ne j$. If the conditional distribution of the linear form $L_2={\beta }_1{\xi }_1+{\beta }_2{\xi }_2+\cdots +{\beta }_n{\xi }_n$ given $L_1={\alpha }_1{\xi }_1+{\alpha }_2{\xi }_2+\cdots +{\alpha }_n{\xi }_n$ is symmetric, then all random variables ${\xi }_j$ are Gaussian. We prove an analogue of this theorem for two independent random variables in the case when they take values in the group of $p$-adic numbers ${\Omega }_p$, and coefficients of linear forms are topological automorphisms of ${\Omega }_p$.

Text overlap in Introduction with arXiv:1103.2617