Abstract
A quasi-infinitely divisible distribution on \(\mathbb {R}^d\) is a probability distribution μ on \(\mathbb {R}^d\) whose characteristic function can be written as the quotient of the characteristic functions of two infinitely divisible distributions on \(\mathbb {R}^d\). Equivalently, it can be characterised as a probability distribution whose characteristic function has a Lévy–Khintchine type representation with a “signed Lévy measure”, a so called quasi–Lévy measure, rather than a Lévy measure. A systematic study of such distributions in the univariate case has been carried out in Lindner, Pan and Sato (Trans Am Math Soc 370:8483–8520, 2018). The goal of the present paper is to collect some known results on multivariate quasi-infinitely divisible distributions and to extend some of the univariate results to the multivariate setting. In particular, conditions for weak convergence, moment and support properties are considered. A special emphasis is put on examples of such distributions and in particular on \(\mathbb {Z}^d\)-valued quasi-infinitely divisible distributions.
In honour of Ron Doney on the occasion of his 80th birthday. This research was supported by DFG grant LI-1026/6-1. Financial support is gratefully acknowledged.
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Notes
- 1.
An answer to this question has been obtained recently by Kutlu [19]. Kutlu shows that \(\mathrm {QID}(\mathbb {R}^d)\) is not dense in \(\mathcal {P}(\mathbb {R}^d)\) if d ≥ 2, by giving an explicit example of a distribution on \(\mathbb {R}^d\) which can not be approximated by quasi-infinitely divisible distributions. In particular, it is shown that its characteristic function can not be approximated arbitrarily well by zero-free continuous functions with respect to uniform convergence on every compact set.
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We would like to thank the anonymous referee for careful reading and valuable suggestions which helped to improve the exposition of the manuscript.
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Berger, D., Kutlu, M., Lindner, A. (2021). On Multivariate Quasi-infinitely Divisible Distributions. In: Chaumont, L., Kyprianou, A.E. (eds) A Lifetime of Excursions Through Random Walks and Lévy Processes. Progress in Probability, vol 78. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-83309-1_6
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