L. Klebanov proved the following theorem. Let \(\xi _1, \dots , \xi _n\) be independent random variables. Consider linear forms \(L_1=a_1\xi _1+\cdots +a_n\xi _n,\) \(L_2=b_1\xi _1+\cdots +b_n\xi _n,\) \(L_3=c_1\xi _1+\cdots +c_n\xi _n,\) \(L_4=d_1\xi _1+\cdots +d_n\xi _n,\) where the coefficients \(a_j, b_j, c_j, d_j\) are real numbers. If the random vectors \((L_1,L_2)\) and \((L_3,L_4)\) are identically distributed, then all \(\xi _i\) for which \(a_id_j-b_ic_j\ne 0\) for all \(j=\overline{1,n}\) are Gaussian random variables. The present article is devoted to an analog of the Klebanov theorem in the case when random variables take values in a locally compact Abelian group and the coefficients of the linear forms are integers.
