Characterization of Probability Distributions on Locally Compact Abelian Groups by the Property of Identical Distribution of Linear Forms with Random Coefficients

Let X be a locally compact Abelian group. We consider linear forms of independent random variables with values in X. In doing so, one of the coefficients of the linear forms is a random variable with a Bernoulli distribution. For some classes of groups we describe possible distributions of random variables provided that the linear forms are identically distributed. The proof of the theorems is reduced to solving some functional equations on the character group of the group X, and to solve functional equations, methods of abstract harmonic analysis are used.