Some positive conclusions related to the Embrechts–Goldie conjecture

We give some conditions under which if an infinitely divisible distribution supported on $[0,\infty)$ belongs to the intersection of the distribution class $\mathcal{L}(\gamma)$ for some $\gamma\ge0$ and the distribution class $\mathcal{OS}$, then so does the corresponding Lévy distribution or its convolution with itself. To this end, we discuss the closure under compound convolution roots for the class and provide some types of distributions satisfying the above conditions. Therefore, this leads to some positive conclusions related to the Embrechts–Goldie conjecture in contrast to the fact that all corresponding previous results for the distribution class $\mathcal{L}(\gamma)\cap\mathcal{OS}$ were negative.