Characterization of probability measures by linear functions defined on a homogeneous Markov chain

Abstract

Let ζ₁, ζ₂, ζ₃ be three independent random variables and Z₁ = ζ₁−ζ₂ and Z₂ = ζ₂−ζ₃. It is known that if the characteristic function of (Z₁, Z₂) does not vanish, then the distribution of (Z₁, Z₂) determines those of ζ₁, ζ₂, ζ₃ up to a possible change in location. Generalizations of this result, to random variables ζ₁,..., ζ_n defined on a homogeneous Markov chain is the sense of Gyires, are obtained.