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 $\mathcal{L}(a^T X)$ for all $a\in \mathbb{N}_0^d$. 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.