On the closure under infinitely divisible distribution roots

For some γ > 0, we show that the distribution class (L(γ) ∩ O S )\ S (γ) is not closed under infinitely divisible distribution roots, that is, we provide examples showing that some infinitely divisible distributions belong to this class but their corresponding Lévy distributions do not. To this end, we explore the structural properties of some distribution classes, give a positive conclusion to the Embrechts–Goldie conjecture, and study some properties of a transformation from a heavy-tailed distribution to a light-tailed one.