On necessary conditions of rational-infinite divisibility for distributions with non-zero discrete parts

2025/09/08 by A. A. Khartov, Khartov, Alexey · 1 citation
Mathematics · #Holomorphic and Operator Theory #Advanced Harmonic Analysis Research #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2509.07244

Abstract

We consider the new class \boldsymbolQ of rational-infinitely (or quasi-infinitely) divisible distribution functions on the real line. By definition, F∈ \boldsymbolQ if there are some infinitely divisible distribution functions F₁ and F₂ such that F₁=F*F₂, where ``*'' is the convolution. The characteristic function of such F admits the Lévy--Khintchine-type representation with a ``signed spectral measure''. The class \boldsymbolQ is a significant extension of the family of infinitely divisible distribution functions and it have already found some applications in several areas. So there is an active interest in this class. In particular, a lot of results have recently appeared on the problem of belonging to the class \boldsymbolQ in terms of characteristic functions. In the paper, we continue this series of results by proposing two necessary conditions for distribution functions from \boldsymbolQ with non-zero discrete parts. Namely, the characteristic functions of such a distribution function and its discrete part are always separated from zero.

Citations

• Statistical Inference for Quasi-Infinitely Divisible Distributions via Fourier Methods
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• Analysis of generalized probability distributions associated with higher\n Landau levels
• Properties of stationary distributions of a sequence of generalized Ornstein-Uhlenbeck processes

Cited by

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

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• On quasi-infinitely divisible distributions