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**On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications.**
*(English)*
Zbl 1054.60012

Heavy-tailed distributions can be used in various insurance and finance applications. The aim of the present paper is to investigate max-sum equivalence and convolution closure properties of certain classes of heavy-tailed distributions. The authors consider the following classes and denotations of heavy-tailed distributions: regular variation \((R)\), consistent variation \((C)\), dominant variation \((D)\), subexponential variation \((S)\), and long-tailed variation \((L)\) distributions. These classes are known to satisfy the following inclusion relations:
\[
R\subset C\subset D\cap L\subset S\subset L.
\]
The main results of the paper consist in extending the properties of the max-sum equivalence and convolution to larger classes of heavy-tailed distributions, and applying these effects to the study of asymptotic behaviour of some important distributions. More precisely, in Section 2, the authors prove that the class \(D\) of heavy-tailed distributions is closed under convolution, and the max-sum equivalence property is shown to hold for the class \(D\cap L\). The class \(C\) is then shown to be closed under convolution. Section 3 considers the closure properties of \(C\) and \(D\cap L\) classes under compound distributions, in particular, under compound geometric distribution. Asymptotic behaviour of the tails of compound geometric convolutions is discussed. These results are applied in Section 4 to the ruin probability in the compound Poisson risk process by an \(\alpha\)-stable Lévy motion, and in Section 5 to the equilibrium waiting-time distribution of the M/G/\(k\) type queue.

Reviewer: Neculai Curteanu (Iaşi)

### MSC:

60E05 | Probability distributions: general theory |

60B15 | Probability measures on groups or semigroups, Fourier transforms, factorization |

91B30 | Risk theory, insurance (MSC2010) |

### Keywords:

max-sum equivalence; convolution closure; heavy-tailed distribution; consistent variation; subexponentiality; long-tailed distribution; compound geometric distribution; \(\alpha\)-stable Lévy process; Poisson risk process; ruin probability; equilibrium waiting-time; M/G/\(k\) queue
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\textit{J. Cai} and \textit{Q. Tang}, J. Appl. Probab. 41, No. 1, 117--130 (2004; Zbl 1054.60012)

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