On a Characterization Theorem on 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 in independent random variables given another. We prove an analogue of this theorem for linear forms in two independent random variables with values in an a-adic solenoind 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 the a-adic solenoid.