A Cramér--Wold device for infinite divisibility of ℤ^d-valued distributions

2020/11/17 by David Berger, Alexander Lindner, Berger, David +1
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #60E07 #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Random Matrices and Applications #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2011.08530

openalex publication_date 2020/11/17 · openalex created_date 2020/11/23 · openalex updated_date 2026/07/28

Abstract

We show that a Cramér--Wold device holds for infinite divisibility of ℤ^d-valued distributions, i.e. that the distribution of a ℤ^d-valued random vector X is infinitely divisible if and only if L(a^T X) is infinitely divisible for all a∈ ℝ^d, and that this in turn is equivalent to infinite divisibility of L(a^T X) for all a∈ ℕ₀^d. A key tool for proving this is a Lévy--Khintchine type representation with a signed Lévy measure for the characteristic function of a ℤ^d-valued distribution, provided the characteristic function is zero-free.

Related

• On multivariate quasi-infinitely divisible distributions
• Zeta distributions generated by Dirichlet series and their (quasi) infinite divisibility
• A criterion and a Cramér-Wold device for quasi-infinite divisibility for discrete multivariate probability laws
• On quasi-infinitely divisible distributions
• On freely quasi-infinitely divisible distributions