A Cram\'er--Wold device for infinite divisibility of ${\mathbb{Z}}^d$-valued distributions

We show that a Cram'er–Wold device holds for infinite divisibility of ${\mathbb{Z}}^d$-valued distributions, i.e. that the distribution of a ${\mathbb{Z}}^d$-valued random vector $X$ is infinitely divisible if and only if $\mathcal{L}(a^{T}X)$ is infinitely divisible for all $a\in {\mathbb{R}}^d$, and that this in turn is equivalent to infinite divisibility of $$\mathcal{L}$(a^T X)$forall$a\in $\mathbb{N}$_0^d$. A key tool for proving this is a L'evy–Khintchine type representation with a signed L'evy measure for the characteristic function of a ${\mathbb{Z}}^d$-valued distribution, provided the characteristic function is zero-free.