We show that a Cramér–Wold device holds for infinite divisibility of -valued distributions, i.e. that the distribution of a -valued random vector X is infinitely divisible if and only if the distribution of is infinitely divisible for all , and that this in turn is equivalent to infinite divisibility of the distribution of for all . A key tool for proving this is a Lévy–Khintchine type representation with a signed Lévy measure for the characteristic function of a -valued distribution, provided the characteristic function is zero-free.
This research was supported by DFG grant LI-1026/6-1. Financial support is gratefully acknowledged.
We would like to thank two anonymous referees, the associate editor and the editor for valuable comments that helped to improve the presentation of the manuscript.
"A Cramér–Wold device for infinite divisibility of -valued distributions." Bernoulli 28 (2) 1276 - 1283, May 2022. https://doi.org/10.3150/21-BEJ1386