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Convolution and convolution-root properties of long-tailed distributions
We obtain a number of new general properties, related to the closedness of
the class of long-tailed distributions under convolutions, that are of interest
themselves and may be applied in many models that deal with "plus" and/or "max"
operations on heavy-tailed random variables. We analyse the closedness property
under convolution roots for these distributions. Namely, we introduce two
classes of heavy-tailed distributions that are not long-tailed and study their
properties. These examples help to provide further insights and, in particular,
to show that the properties to be both long-tailed and so-called "generalised
subexponential" are not preserved under the convolution roots. This leads to a
negative answer to a conjecture of Embrechts and Goldie [10, 12] for the class
of long-tailed and generalised subexponential distributions. In particular, our
examples show that the following is possible: an infinitely divisible
distribution belongs to both classes, while its Levy measure is neither
long-tailed nor generalised subexponential.Comment: 21page
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