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 $a_id_j-b_ic_j\neq 0$ 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.

翻译：L. Klebanov 证明了以下定理：设 ξ₁, …, ξ_n 为独立随机变量。考虑线性形式 L₁ = a₁ξ₁ + … + a_nξ_n，L₂ = b₁ξ₁ + … + b_nξ_n，L₃ = c₁ξ₁ + … + c_nξ_n，L₄ = d₁ξ₁ + … + d_nξ_n，其中系数 a_j, b_j, c_j, d_j 为实数。若随机向量 (L₁, L₂) 与 (L₃, L₄) 同分布，则所有满足对任意 j=1,…,n 有 a_id_j - b_ic_j ≠ 0 的 ξ_i 均为高斯随机变量。本文致力于研究当随机变量取值于局部紧阿贝尔群且线性形式系数为整数时 Klebanov 定理的类比结果。