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We consider questions related to the well-known conjecture due to Embrechtsand Goldie on the closedness of different classes of heavy- and light-taileddistributions with respect to convolution roots. We show that the classL(\gamma)\cap OS is not closed under convolution roots related to an infinitelydivisible distribution for any \gamma\ge0, i.e. we provide examples ofinfinitely divisible distributions belonging to this class such that thecorresponding Levy spectral distribution does not. We also prove a similarstatement for the class (L(\gamma)\cap OS)\ S(\gamma). In order to facilitateour analysis, we explore the structural properties of some of the classes ofdistributions, and study some properties of the well-known transformation froma heavy-tailed distribution to a light-tailed one.

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