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

Abstract

We show that a Cramér--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 Lévy--Khintchine type representation with a signed Lévy measure for the characteristic function of a ${\mathbb{Z}}^d$-valued distribution, provided the characteristic function is zero-free.