Convolution equivalence and distributions of random sums

Abstract

A serious gap in the Proof of Pakes's paper on the convolution equivalence of infinitely divisible distributions on the line is completely closed. It completes the real analytic approach to Sgibnev's theorem. Then the convolution equivalence of random sums of IID random variables is discussed. Some of the results are applied to random walks and Lévy processes. In particular, results of Bertoin and Doney and of Korshunov on the distribution tail of the supremum of a random walk are improved. Finally, an extension of Rogozin's theorem is proved.

Convolution Equivalence and Infinite Divisibility: Corrections and Corollaries

“…Watanabe (2007) proves that Theorem 3.3(c) implies Theorem 3.3(b) (and hence (a)) by using a clever adaptation of the first part of the proof of Theorem 4.2 of Embrechts and Goldie (1982). Theorem 3.3 as stated (i.e.…”

Reinsurance under the LCR and ECOMOR treaties with emphasis on light-tailed claims

“…Recent studies of this class can be found in Pakes (2004), Tang (2006), Foss and Korshunov (2007), and Watanabe (2008), among many others. This class is often used to model claim-size distributions; see, for example, Embrechts and Veraverbeke (1982), Klüppelberg (1989a), and Tang and Tsitsiashvili (2004).…”

Extremes of Lévy driven mixed MA processes with convolution equivalent distributions

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