Independent Linear Statistics on Finite Abelian Groups

Abstract

We give a complete description of the class of all finite Abelian groups X for which the independence of linear statistics L ₁ = α₁(ξ₁) + α₂(ξ₂) + α₃(ξ₃) and L ₂ = β₁(ξ₁) + β₂(ξ₂) + β₃(ξ₃) (here, ξ _j , j = 1, 2, 3, are independent random variables with values in X and distributions μ _j ; α _j and β _j are automorphisms of X) implies that either one, or two, or three of the distributions μ _j are idempotents.