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The class $\boldsymbol{Q}$ and mixture distributions with dominated continuous singular parts

A. A. Khartov

TL;DR

The problem is to extend the theory of quasi-infinitely divisible distributions to include a continuous singular part under a domination condition by the discrete part. The approach yields a Lévy-type representation with a signed spectral measure and a potentially nontrivial continuous singular component encoded by W in V, combining discrete, absolutely continuous, and continuous singular elements in the Lévy triplet L = Ld + La + Ls. Key contributions include a general domination-based criterion for membership in Q, a constructive form for the characteristic function, and a positive solution to Lindner-Pan-Sato's decomposition question in this regime. The results broaden quasi-infinitely divisible theory to mixed distributions and provide tools for models with Cantor-type singular parts and related limit theorems.

Abstract

We consider a new class $\boldsymbol{Q}$ of distribution functions $F$ that have the property of rational-infinite divisibility: there exist some infinitely divisible distribution functions $F_1$ and $F_2$ such that $F_1=F*F_2$. A distribution function of the class $\boldsymbol{Q}$ is quasi-infinitely divisible in the sense that its characteristic function admits the Lévy-type representation with a ``signed spectral measure''. This class is a wide natural extension of the fundamental class of infinitely divisible distribution functions and it is actively studied now. We are interested in conditions for a distribution function $F$ to belong to the class $\boldsymbol{Q}$ for the unexplored case, where $F$ may have a continuous singular part. We propose a criterion under the assumption that the continuous singular part of $F$ is dominated by the discrete part in a certain sense. The criterion generalizes the previous results by Alexeev and Khartov for discrete probability laws and the results by Berger and Kutlu for the mixtures of discrete and absolutely continuous laws. In addition, we describe the characteristic triplet of the corresponding Lévy-type representation, which may contain some continuous singular part. We also show that the assumption of the dominated continuous singular part cannot be simply omitted or even slightly extended (without some special assumptions). We apply the general criterion to some interesting particular examples. We also positively solve the decomposition problem stated by Lindner, Pan and Sato within the considered case.

The class $\boldsymbol{Q}$ and mixture distributions with dominated continuous singular parts

TL;DR

The problem is to extend the theory of quasi-infinitely divisible distributions to include a continuous singular part under a domination condition by the discrete part. The approach yields a Lévy-type representation with a signed spectral measure and a potentially nontrivial continuous singular component encoded by W in V, combining discrete, absolutely continuous, and continuous singular elements in the Lévy triplet L = Ld + La + Ls. Key contributions include a general domination-based criterion for membership in Q, a constructive form for the characteristic function, and a positive solution to Lindner-Pan-Sato's decomposition question in this regime. The results broaden quasi-infinitely divisible theory to mixed distributions and provide tools for models with Cantor-type singular parts and related limit theorems.

Abstract

We consider a new class of distribution functions that have the property of rational-infinite divisibility: there exist some infinitely divisible distribution functions and such that . A distribution function of the class is quasi-infinitely divisible in the sense that its characteristic function admits the Lévy-type representation with a ``signed spectral measure''. This class is a wide natural extension of the fundamental class of infinitely divisible distribution functions and it is actively studied now. We are interested in conditions for a distribution function to belong to the class for the unexplored case, where may have a continuous singular part. We propose a criterion under the assumption that the continuous singular part of is dominated by the discrete part in a certain sense. The criterion generalizes the previous results by Alexeev and Khartov for discrete probability laws and the results by Berger and Kutlu for the mixtures of discrete and absolutely continuous laws. In addition, we describe the characteristic triplet of the corresponding Lévy-type representation, which may contain some continuous singular part. We also show that the assumption of the dominated continuous singular part cannot be simply omitted or even slightly extended (without some special assumptions). We apply the general criterion to some interesting particular examples. We also positively solve the decomposition problem stated by Lindner, Pan and Sato within the considered case.
Paper Structure (4 sections, 14 theorems, 197 equations)

This paper contains 4 sections, 14 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Criteria for belonging to class $\boldsymbol{Q}$
  3. Auxiliary theorems and lemmata
  4. Proofs

Key Result

Theorem 1

Suppose that $F$ is a discrete distribution function, $c_d=1$ and $c_a=c_s=0$ in repr_F_Lebesgue$($$F$ and $F_d$ are identical and hence $f$ and $f_d$ are too; $F$ has form def_Fd, $f$ is represented by def_fd$)$. Then $F\in\boldsymbol{Q}$ if and only if $\inf_{t\in\mathbb R} |f(t)| > 0$. In that ca with some $\gamma_0 \in \langle \mathcal{X} \rangle$ and $\lambda_u \in \mathbb R$ for all $u \in \

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Proposition 1
  • Remark 2
  • Remark 3
  • Proposition 2
  • Proposition 3
  • Theorem 4
  • ...and 8 more