On decomposition problem for distribution functions of class \boldsymbolQ

2024/12/25 by Khartov, A. A.
#60E05 #60E07 #60E10 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2412.18915

Abstract

We consider a new class \boldsymbolQ of distribution functions F that have the property of rational-infinite divisibility: there exist some infinitely divisible distribution functions F₁ and F₂ such that F₁=F*F₂. A distribution function of the class \boldsymbolQ is quasi-infinitely divisible in the sense that its characteristic function admits the Lévy--Khinchine type representation with a ``signed spectral measure''. The class \boldsymbolQ, being a natural extension of the class \boldsymbolI of infinitely divisible distribution functions, is actively studied now and it finds various applications. In 2018, Lindner, Pan and Sato formulated the open question: is it true that if F∈\boldsymbolQ and F=F₁*F₂ with some distribution functions F₁ and F₂, then F₁∈\boldsymbolQ and F₂∈\boldsymbolQ? There are some positive results under special assumptions on the type of F. In this paper, we answer the question in a general setting without any additional assumptions. We also consider the same question but with the stronger assumption that F∈\boldsymbolI.

Related

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