Some positive conclusions related to the Embrechts-Goldie' conjecture

In this paper, we give some conditions, under which, if an infinitely divisible distribution supported on $[0,\infty)$ belongs to the intersection of exponential distribution class $\mathcal{L}(\gamma)$ for some $\gamma\ge0$ and generalised subexponential distribution class $\mathcal{OS}$, then its L$\rm\acute{e}$vy spectral distribution or convolution of the distribution with itself also belongs to the same one. To this end, we discuss the closure under the compound convolution roots for the class. In addition, we do some in-depth discussion about the above-mentioned conditions, and provide some types of distributions satisfying them. Further, we obtain some local versions of the above-mentioned results by the Esscher transform of distributions. Therefore, some positive conclusions related to the Embrechts-Goldie conjecture are obtained. Prior to this, all corresponding results are negative