May 2022 A Cramér–Wold device for infinite divisibility of Zd-valued distributions
David Berger, Alexander Lindner
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Bernoulli 28(2): 1276-1283 (May 2022). DOI: 10.3150/21-BEJ1386


We show that a Cramér–Wold device holds for infinite divisibility of Zd-valued distributions, i.e. that the distribution of a Zd-valued random vector X is infinitely divisible if and only if the distribution of aTX is infinitely divisible for all aRd, and that this in turn is equivalent to infinite divisibility of the distribution of aTX for all aN0d. A key tool for proving this is a Lévy–Khintchine type representation with a signed Lévy measure for the characteristic function of a Zd-valued distribution, provided the characteristic function is zero-free.

Funding Statement

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.


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David Berger. Alexander Lindner. "A Cramér–Wold device for infinite divisibility of Zd-valued distributions." Bernoulli 28 (2) 1276 - 1283, May 2022.


Received: 1 November 2020; Revised: 1 March 2021; Published: May 2022
First available in Project Euclid: 3 March 2022

MathSciNet: MR4388938
zbMATH: 07526584
Digital Object Identifier: 10.3150/21-BEJ1386

Keywords: Cramér–Wold device , infinitely divisible distribution , quasi-infinitely divisible distribution , signed Lévy measure

Rights: Copyright © 2022 ISI/BS


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Vol.28 • No. 2 • May 2022
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