On closedness under convolution roots related to an infinitely divisible distribution in the distribution class L(γ)

2015/12/06 by Hui Xu, Xu, Hui, Yuebao Wang +5 · 1 citation
Computer Science · Economics, Econometrics and Finance · Mathematics · #60E05 #60F10 #60G50 #Advanced Harmonic Analysis Research #Bayesian Methods and Mixture Models #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #math.PR #msc:60E05 #msc:60F10 #msc:60G50

paper · pdf · doi:10.48550/arxiv.1512.01792

28 pages

openalex publication_date 2015/12/06 · openalex created_date 2016/06/24 · arxiv created 2016/09/27 · arxiv updated 2016/09/28 · openalex updated_date 2026/07/28

Abstract

We consider questions related to the well-known conjecture due to Embrechts and Goldie on the closedness of different classes of heavy- and light-tailed distributions with respect to convolution roots. We show that the class L(γ)∩ OS is not closed under convolution roots related to an infinitely divisible distribution for any γ≥0, i.e. we provide examples of infinitely divisible distributions belonging to this class such that the corresponding Levy spectral distribution does not. We also prove a similar statement for the class (L(γ)∩ OS) S(γ). In order to facilitate our analysis, we explore the structural properties of some of the classes of distributions, and study some properties of the well-known transformation from a heavy-tailed distribution to a light-tailed one.

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