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

Abstract

We show that a Cramér–Wold device holds for infinite divisibility of ${\mathbb{Z}}^{\mathit{d}}$-valued distributions, i.e. that the distribution of a ${\mathbb{Z}}^{\mathit{d}}$-valued random vector X is infinitely divisible if and only if the distribution of ${\mathit{a}}^{\mathit{T}}\mathit{X}$ is infinitely divisible for all $\mathit{a}\in {\mathbb{R}}^{\mathit{d}}$, and that this in turn is equivalent to infinite divisibility of the distribution of ${\mathit{a}}^{\mathit{T}}\mathit{X}$ for all $\mathit{a}\in {\mathbb{N}}_0^{\mathit{d}}$. A key tool for proving this is a Lévy–Khintchine type representation with a signed Lévy measure for the characteristic function of a ${\mathbb{Z}}^{\mathit{d}}$-valued distribution, provided the characteristic function is zero-free.