Title: A characterization of signed discrete infinitely divisible 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 Lévy-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.