Simple conditions for mixing of infinitely divisible processes

Let (Xt)tϵT be a real-valued, stationary, infinitely divisible stochastic process. We show that (Xt)tϵT is mixing if and only if Eei(Xt − X0) → |EeiX0|2, provided the Lévy measure of X0 has no atoms in 2πZ. We also show that if (Xt)tϵT is given by a stochastic integral with respect to an infinitely divisible measure then the mixing of (Xt)tϵT is equivalent to the essential disjointness of the supports of the representing functions.