On quasi-infinitely divisible distributions

TL;DR

This paper explores the properties of quasi-infinitely divisible distributions, which generalize infinitely divisible distributions by allowing a signed Lévy measure, and demonstrates their density and characterizations on the real line.

Contribution

It provides a detailed analysis of quasi-infinitely divisible distributions, including their support, moments, and convergence properties, and characterizes those on the integers via their characteristic functions.

Findings

Quasi-infinitely divisible distributions are dense in all probability distributions.

On the integers, such distributions are characterized by non-zero characteristic functions.

The set of quasi-infinitely divisible distributions includes all distributions with non-vanishing characteristic functions on the integers.

Abstract

A quasi-infinitely divisible distribution on $\mathbb{R}$ is a probability distribution whose characteristic function allows a L\'evy-Khintchine type representation with a "signed L\'evy measure", rather than a L\'evy measure. Quasi-infinitely divisible distributions appear naturally in the factorization of infinitely divisible distributions. Namely, a distribution $\mu$ is quasi-infinitely divisible if and only if there are two infinitely divisible distributions $\mu_1$ and $\mu_2$ such that $\mu_1 \ast \mu = \mu_2$. The present paper studies certain properties of quasi-infinitely divisible distributions in terms of their characteristic triplet, such as properties of supports, finiteness of moments, continuity properties and weak convergence, with various examples constructed. In particular, it is shown that the set of quasi-infinitely divisible distributions is dense in the set of all probability distributions with respect to weak convergence. Further, it is proved that a distribution concentrated on the integers is quasi-infinitely divisible if and only if its characteristic function does not have zeroes, with the use of the Wiener-L\'evy theorem on absolutely convergent Fourier series. A number of fine properties of such distributions are proved based on this fact. A similar characterisation is not true for non-lattice probability distributions on the line.