Convolution and convolution-root properties of long-tailed distributions

Hui Xu, Serguei Foss, Yuebao Wang

Research output: Contribution to journalArticle

15 Citations (Scopus)
59 Downloads (Pure)

Abstract

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 (J. Austral. Math. Soc. (Ser. A) 29, 243–256 1980, Stoch. Process. Appl. 13, 263–278 1982) 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 Lévy measure is neither long-tailed nor generalised subexponential.
Original languageEnglish
Pages (from-to)605-628
Number of pages24
JournalExtremes
Volume18
Issue number4
Early online date24 Sep 2015
DOIs
Publication statusPublished - Dec 2015

Keywords

  • Closedness
  • Convolution
  • Convolution root
  • Generalised subexponential distribution
  • Infinitely divisible distribution
  • Long-Tailed distribution
  • Lévy measure
  • Random sum

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