On multivariate quasi-infinitely divisible distributions

2021/01/07 by Berger, David, Kutlu, Merve, Lindner, Alexander · 2 citations
#60E07 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2101.02544

Abstract

A quasi-infinitely divisible distribution on ℝ^d is a probability distribution μ on ℝ^d whose characteristic function can be written as the quotient of the characteristic functions of two infinitely divisible distributions on 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 \citelindner. 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 ℤ^d-valued quasi-infinitely divisible distributions.

Cited by

• Denseness in total variation and the class of rational-infinitely divisible distributions
• On necessary conditions of rational-infinite divisibility for distributions with non-zero discrete parts

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