On the Heyde Theorem for Finite Abelian Groups

Abstract

It is well-known Heyde's characterization theorem for the Gaussian distribution on the real line: if ξ _j are independent random variables, α _j , β _j are nonzero constants such that β _i α ±β _j α⁻¹ _j ≠ 0 for all i≠ j and the conditional distribution of L ₂=β₁ ξ₁ + ··· + β _n ξ _n given L ₁=α₁ ξ₁ + ··· +α _n ξ _n is symmetric, then all random variables ξ _j are Gaussian. We prove some analogs of this theorem, assuming that independent random variables take on values in a finite Abelian group X and the coefficients α _j ,β _j are automorphisms of X.