Simple conditions for mixing of infinitely divisible processes

Let (Xt)t[epsilon]T be a real-valued, stationary, infinitely divisible stochastic process. We show that (Xt)t[epsilon]T is mixing if and only if Eei(Xt - X0) --> EeiX02, provided the Lévy measure of X0 has no atoms in 2[pi]Z. We also show that if (Xt)t[epsilon]T is given by a stochastic integral with respect to an infinitely divisible measure then the mixing of (Xt)t[epsilon]T is equivalent to the essential disjointness of the supports of the representing functions.