An Analogue of the Klebanov Theorem for Locally Compact Abelian Groups

L. Klebanov proved the following theorem. Let $$\xi _1, \dots , \xi _n$$ ξ 1 , ⋯ , ξ n be independent random variables. Consider linear forms $$L_1=a_1\xi _1+\cdots +a_n\xi _n,$$ L 1 = a 1 ξ 1 + ⋯ + a n ξ n , $$L_2=b_1\xi _1+\cdots +b_n\xi _n,$$ L 2 = b 1 ξ 1 + ⋯ + b n ξ n , $$L_3=c_1\xi _1+\cdots +c_n\xi _n,$$ L 3 = c 1 ξ 1 + ⋯ + c n ξ n , $$L_4=d_1\xi _1+\cdots +d_n\xi _n,$$ L 4 = d 1 ξ 1 + ⋯ + d n ξ n , where the coefficients $$a_j, b_j, c_j, d_j$$ a j , b j , c j , d j are real numbers. If the random vectors $$(L_1,L_2)$$ ( L 1 , L 2 ) and $$(L_3,L_4)$$ ( L 3 , L 4 ) are identically distributed, then all $$\xi _i$$ ξ i for which $$a_id_j-b_ic_j\ne 0$$ a i d j - b i c j ≠ 0 for all $$j=\overline{1,n}$$ j = 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.