Identically distributed random vectors on locally compact Abelian groups

08/10/2023

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by   Margaryta Myronyuk, et al.

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L. Klebanov proved the following theorem. Let ξ_1, …, ξ_n be independent random variables. Consider linear forms L_1=a_1ξ_1+⋯+a_nξ_n, L_2=b_1ξ_1+⋯+b_nξ_n, L_3=c_1ξ_1+⋯+c_nξ_n, L_4=d_1ξ_1+⋯+d_nξ_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 ξ_i for which a_id_j-b_ic_j≠ 0 for all 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.