A Cramér–Wold device for infinite divisibility of Zd-valued distributions

Abstract

We show that a Cramér–Wold device holds for infinite divisibility of Zd-valued distributions, i.e. that the distribution of a Zd-valued random vector X is infinitely divisible if and only if the distribution of aTX is infinitely divisible for all a∈Rd, and that this in turn is equivalent to infinite divisibility of the distribution of aTX for all a∈N0d. A key tool for proving this is a Lévy–Khintchine type representation with a signed Lévy measure for the characteristic function of a Zd-valued distribution, provided the characteristic function is zero-free.