Simple conditions for mixing of infinitely divisible processes

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Abstract

Let (Xt)T be a real-valued, stationary, infinitely divisible stochastic process. We show that (Xt)T is mixing if and only if Eei(XtX0) → |EeiX0|2, provided the Lévy measure of X0 has no atoms in 2πZ. We also show that if (Xt)T is given by a stochastic integral with respect to an infinitely divisible measure then the mixing of (Xt)T is equivalent to the essential disjointness of the supports of the representing functions.

Keywords

Stationary process
Infinitely divisible process
Mixing
Weak mixing

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The research of the first named author was supported in part by NSF Grant DMS-9406294 and the Tennessee Science Alliance; the research of the second named author was supported in part by KBN Grant and the Tennessee Science Alliance.