We obtain a number of new general properties, related to the closedness ofthe class of long-tailed distributions under convolutions, that are of interestthemselves and may be applied in many models that deal with "plus" and/or "max"operations on heavy-tailed random variables. We analyse the closedness propertyunder convolution roots for these distributions. Namely, we introduce twoclasses of heavy-tailed distributions that are not long-tailed and study theirproperties. These examples help to provide further insights and, in particular,to show that the properties to be both long-tailed and so-called "generalisedsubexponential" are not preserved under the convolution roots. This leads to anegative answer to a conjecture of Embrechts and Goldie [10, 12] for the classof long-tailed and generalised subexponential distributions. In particular, ourexamples show that the following is possible: an infinitely divisibledistribution belongs to both classes, while its Levy measure is neitherlong-tailed nor generalised subexponential.
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