On a characterisation theorem for $a$-adic solenoids

Abstract

According to the Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given another. We prove an analogue of this theorem for linear forms of two independent random variables taking values in an $a$-adic solenoid $\Sigma_a$ without elements of order 2, assuming that the characteristic functions of the random variables do not vanish, and coefficients of the linear forms are topological automorphisms of $\Sigma_a$.