The Klebanov theorem is a characteristic theorem that considers the equal distribution of two pairs of linear forms, that is using four arbitrary linear forms of independent random variables with real values. The theorem is stated as follows:
Let \(\xi_1, \dots,\xi_n\) be independent random variables. Consider linear forms \(L_1 = a_1 \xi_1+ \dots+a_n \xi_n\), \(L_2 = b_1 \xi_1+ \dots+b_n \xi_n\), \(L_3 = c_1 \xi_1+ \dots+c_n \xi_n\), \(L_4 = d_1 \xi_1+ \dots+d_n \xi_n\), where the coefficients \(a_j, b_j, c_j, d_j\), are real numbers. If the random variable \((L_1, L_2)\) and \((L_3, L_4)\) are identically distributed, then all the \(\xi_i\) for which \(ajdj - bjcj \neq 0\) for all \(j= 1, .. n\) are Gaussian random variambles.
In the article the author considers the Klebanov theorem when the random variables take values on a second countable locally compact group and the coefficients of the linear forms are intergers. In the process the author establishes necessary results to provide the analogue result for the group case, for instance he establishes and analogue to the Cramer theorem on decomposition and the Marcinkiewicz theorem.