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 $ξ_1, ξ_2,\dots, ξ_n$, $n\ge 2,$ be independent random variables, let $α_j, β_j$ be nonzero constants such that $β_iα_i^{-1} + β_jα_j^{-1} \ne 0$ for all $i \ne j$. If the conditional distribution of the linear form $L_2 = β_1ξ_1 + β_2ξ_2+ \cdots + β_nξ_n$ given $L_1 = α_1ξ_1 + α_2ξ_2+\cdots + α_nξ_n$ is symmetric, then all random variables $ξ_j$ 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 $Ω_p$, and coefficients of linear forms are topological automorphisms of $Ω_p$.