On the Skitovich-Darmois Theorem on Abelian Groups

We prove theorems that generalize the Skitovich-Darmois theorem to the case where independent random variables ξ_j, j = 1, 2, ..., n, n ≥ 2, take values in a locally compact Abelian group and the coefficients α_j and β_j of the linear forms L ₁ = α₁ξ₁ + ... + α_nξ_n and L ₂ = β₁ξ₁ + ... + β_nξ_n are automorphisms of this group.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 10, pp. 1342 – 1356, October, 2004.