Closure under infinitely divisible distribution roots and the Embrechts–Goldie conjecture

We show that the distribution class ℒ(γ) \ 𝒪𝒮 is not closed under infinitely divisible distribution roots for γ > 0, that is, we provide some infinitely divisible distributions belonging to the class, whereas the corresponding Lévy distributions do not. In fact, one part of these Lévy distributions belonging to the class 𝒪ℒ\ℒ(γ) have different properties, and the other parts even do not belong to the class 𝒪ℒ. Therefore, combining with the existing related results, we give a completely negative conclusion for the subject and Embrechts–Goldie conjecture. Then we discuss some interesting issues related to the results of this paper.