# A characterization of signed discrete infinitely divisible distributions

## TL;DR

This paper extends Feller's characterization to signed discrete infinitely divisible distributions, connecting them with DPCP distributions and addressing an open problem in the field.

## Contribution

It introduces a characterization of signed discrete infinitely divisible distributions, linking them to DPCP and solving an open problem posed by Sato, Chaumont, and Yor.

## Key findings

- Many distributions, like mixed Poisson, are DPCP.
- DPCP distributions have unique properties, such as a lower bound on the parameter .
- An analogous characteristic function result is established for signed integer-valued distributions.

## Abstract

In this article, we give some reviews concerning negative probabilities model and quasi-infinitely divisible at the beginning. We next extend Feller's characterization of discrete infinitely divisible distributions to signed discrete infinitely divisible distributions, which are discrete pseudo compound Poisson (DPCP) distributions with connections to the L\'evy-Wiener theorem. This is a special case of an open problem which is proposed by Sato(2014), Chaumont and Yor(2012). An analogous result involving characteristic functions is shown for signed integer-valued infinitely divisible distributions. We show that many distributions are DPCP by the non-zero p.g.f. property, such as the mixed Poisson distribution and fractional Poisson process. DPCP has some bizarre properties, and one is that the parameter $\lambda $ in the DPCP class cannot be arbitrarily small.