Convolution and convolution-root properties of long-tailed distributions

Hui Xu, Sergey Foss, Yuebao Wang

Published 2015-01-29Version 1

We obtain a number of new general convolution properties related to long-tailed distributions. Then we show that the properties to be either long-tailed or generalised subexponential are not preserved under the convolution roots. Namely, we introduce two families of distributions such that each distribution is neither long-tailed nor generalised subexponential while its n-fold convolution is both long-tailed and generalised subexponential, for all n>=2. This leads to the negative answer to the conjecture of Embrechts and Goldie [10, 12] in the class of long-tailed and generalised subexponential distributions. Further, we analyse corresponding problems for distributions of random sums of random variables. In particular, we provide an example of a long-tailed and generalised subexponential infinitely divisible distribution whose Levy measure does not have these properties.