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

2025/10/28 by Khartov, Alexey
#60E05 #60E07 #60E10 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2510.24219

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 Lévy--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.

Citations

• On necessary conditions of rational-infinite divisibility for distributions with non-zero discrete parts
• Some criteria of rational-infinite divisibility for probability laws
• Quasi-infinite divisibility of a class of distributions with discrete part
• A criterion of quasi-infinite divisibility for discrete laws
• On a denseness result for quasi-infinitely divisible distributions
• Spectral representations of characteristic functions of discrete probability laws
• On multivariate quasi-infinitely divisible distributions
• On Quasi-Infinitely Divisible Distributions with a Point Mass
• On quasi-infinitely divisible distributions

Related

• On necessary conditions of rational-infinite divisibility for distributions with non-zero discrete parts
• On quasi-infinitely divisible distributions
• On a denseness result for quasi-infinitely divisible distributions
• The class \boldsymbolQ and mixture distributions with dominated continuous singular parts
• On freely quasi-infinitely divisible distributions