Denseness in total variation and the class of rational-infinitely divisible distributions

TL;DR

This paper introduces and analyzes a new class of rational-infinitely divisible distributions, exploring their properties, density in probability laws, and convergence behaviors in total variation.

Contribution

It defines the class of rational-infinitely divisible distributions, studies their properties, and investigates their denseness in total variation among probability laws.

Findings

The class is dense in all probability laws under weak convergence.

The class exhibits varied denseness properties depending on the support of distributions.

Results include both positive and negative findings on total variation convergence.

Abstract

We study a new class of so-called rational-infinitely (or quasi-infinitely) divisible probability laws on the real line. The characteristic functions of these distributions are ratios of the characteristic functions of classical infinitely divisible laws and they admit L\'evy--Khinchine type representations with ``signed spectral measures''. This class is rather wide and it has a lot of nice properties. For instance, this class is dense in the family of all (univariate) probability laws with respect to weak convergence. In this paper, we consider the questions concerning a denseness of this class with respect to convergence in total variation. The problem is considered separately for different types of probability laws taking into account the supports of the distributions. A series of ``positive'' and ``negative'' results are obtained.