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

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.