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 $\Omega_p$.