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Convolution and convolution-root properties of long-tailed distributions

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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.

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Authors and Affiliations

  1. School of Mathematical Sciences, Soochow University, Suzhou, 215006, China

    Hui Xu & Yuebao Wang

  2. School of MACS and Maxwell Institute, Heriot-Watt University, Edinburgh, EH14 4AS, UK

    Sergey Foss

  3. Sobolev’s Institute of Mathematics and Novosibirsk State University, Novosibirsk State University and Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia

    Sergey Foss

Authors
  1. Hui Xu
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  2. Sergey Foss
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  3. Yuebao Wang
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Corresponding author

Correspondence to Yuebao Wang.

Additional information

Sergey Foss, Research supported by EPSRC grant No. EP/I017054/1

Yuebao Wang, Research supported by National Science Foundation of China, Grant No.11071182

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Xu, H., Foss, S. & Wang, Y. Convolution and convolution-root properties of long-tailed distributions. Extremes 18, 605–628 (2015). https://doi.org/10.1007/s10687-015-0224-2

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  • DOI: https://doi.org/10.1007/s10687-015-0224-2

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