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

It is well known Heyde's characterization of the Gaussian distribution on the real line: Let, xi(1), xi(2), ..., xi(n), n >= 2, be independent random variables, alpha(j), beta(j) be nonzero constants such that beta(i)alpha(-1)(i) + beta(j)alpha(-1)(j) not equal 0 for all i not equal j. If the conditional distribution of the linear form L-2 = beta(1)xi(1) + beta(2)xi(2) + ... + beta(n)xi(n) given L-1 = alpha(1)xi(1) + alpha(2)xi(2) + ... + 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),.