On a characterization theorem for the group of \(p\)-adic numbers (Q2834178)

\textit{C. C. Heyde}'s theorem [Sankhyā, Ser. A 32, 115--118 (1970; Zbl 0209.50702)] states that if \(\xi_1,\dots, \xi_n\) are independent random variables, if \(\alpha_j\), \(\beta_j\) are non-zero constants such that \(\beta_i\,\alpha_j^{-1}+\beta_j\,\alpha_i^{-1}\neq 0\) when \(i\neq j\) and if the conditional distribution of the linear form \(L_2=\sum_i \beta_i\,\xi_i\) given \(L_1=\sum_i \alpha_j\,\xi_j\) is symmetric, then all the \(\xi_i\)s are Gaussian.NEWLINENEWLINE The author proves an analogue (see Theorem 1) of this characterisation result for two independent random variables \(\xi_1\) and \(\xi_2\) when these latter ones take values in the group \(\Omega_p\) of \(p\)-adic numbers and the coefficients of the linear forms are topological automorphisms of \(\Omega_p\). The whole paper is devoted to the proof of the above theorem in the course of which several functional equations are solved.