## 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.

### MSC:

 60E05 Probability distributions: general theory 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 91B30 Risk theory, insurance (MSC2010)
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### References:

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