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

Alexey Khartov

TL;DR

This work investigates the denseness in total variation of the class $\boldsymbol{Q}$ of rational-infinitely divisible distributions, a broad extension of infinite divisibility realized as ratios of characteristic functions with a Lévy–Khinchine type form and signed spectral measures. It provides constructive positive results showing total-variation denseness for absolutely continuous and discrete lattice distributions (and their mixtures), using a key lemma that ensures nonvanishing of certain convex combinations of characteristic functions and convolution with simple perturbation distributions. Conversely, it establishes negative results for discrete non-lattice and, in fact, for the full class $\boldsymbol{Q}$, demonstrating that total variation density fails in these cases via carefully designed counterexamples and zero-avoidance arguments. Overall, the paper delineates when quasi-infinitely divisible laws can approximate targets in the stronger total-variation metric, clarifying both the capabilities and limits of $\boldsymbol{Q}$ in strong probabilistic topologies.

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.

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

TL;DR

This work investigates the denseness in total variation of the class of rational-infinitely divisible distributions, a broad extension of infinite divisibility realized as ratios of characteristic functions with a Lévy–Khinchine type form and signed spectral measures. It provides constructive positive results showing total-variation denseness for absolutely continuous and discrete lattice distributions (and their mixtures), using a key lemma that ensures nonvanishing of certain convex combinations of characteristic functions and convolution with simple perturbation distributions. Conversely, it establishes negative results for discrete non-lattice and, in fact, for the full class , demonstrating that total variation density fails in these cases via carefully designed counterexamples and zero-avoidance arguments. Overall, the paper delineates when quasi-infinitely divisible laws can approximate targets in the stronger total-variation metric, clarifying both the capabilities and limits of in strong probabilistic topologies.

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.

Paper Structure

This paper contains 4 sections, 13 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. The main tools
  3. The "positive" results
  4. The "negative" results

Key Result

Lemma 1

Let $F_0$ be a distribution function on $\mathbb R$ with a bounded $\mathcal{S}(F_0)$. Let $f_0$ be the characteristic function of $F_0$. Let $\gamma_0$ be an arbitrary real number such that $\gamma_0\ne \mathrm{cext}(F_0)$ if $F_0$ is shift-symmetric. Then for all $\delta\in(0,1)$ except a countabl In particular, for any $\tau\in(0,1]$ there exists $\delta\in(0,\tau)$ such that lm_key_conc holds.

Theorems & Definitions (13)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • Theorem 3
  • Corollary 3
  • Theorem 4
  • ...and 3 more