Asymptotics in the symmetrization inequality

Geluk, Jaap

doi:10.1016/j.spl.2004.06.004

Cited by 6 publications

(5 citation statements)

References 15 publications

(9 reference statements)

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“…The defining relation is again (1.4). Subexponential distributions on the real line were studied before by Willekens [40], Omey [31] and Geluk [16].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Weighted sums of subexponential random variables and asymptotic dependence between returns on reinsurance equities

Geluk

Vries

2006

Insurance: Mathematics and Economics

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Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. ABSTRACT. Suppose X 1 , X 2 , . . . are independent subexponential random variables with partial sums S n . We show that if the pairwise sums of the X i 's are subexponential, then S n is subexponential and Terms of use: Documents inThe result is applied to give conditions under which P (Asymptotic tail probabilities for bivariate linear combinations of subexponential random variables are given. These results are applied to explain the joint movements of the stocks of reinsurers. Portfolio investment and retrocession practices in the reinsurance industry, for reasons of diversification, exposes different reinsurers to the same risks on both sides of their balance sheets. Assuming, in line with the industry practice that the risk drivers follow subexponential distributions, we derive (under mild conditions) when the reinsurer's equity returns are asymptotically dependent, exposing the industry to systemic risk.

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“…The defining relation is again (1.4). Subexponential distributions on the real line were studied before by Willekens [40], Omey [31] and Geluk [16].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The first Lemma (see [16]) shows that it is sufficient to study the behaviour of nonnegative random variables under maxima and convolutions. The next Lemma is well known (see e.g.…”

Section: Weighted Sums Of Iid Random Variables In Smentioning

confidence: 99%

Weighted sums of subexponential random variables and asymptotic dependence between returns on reinsurance equities

Geluk

Vries

2006

Insurance: Mathematics and Economics

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“…Part (a) is a combination of Lemma 1 of Geluk [12] and Lemma 4.2 of Ng et al [25], while part (b) can be found in Embrechts and Goldie [8]. To prove part (c), note that, by part (a), the relation…”

Section: Heavy-tailed Distributionsmentioning

confidence: 88%

Asymptotic Tail Probabilities of Sums of Dependent Subexponential Random Variables

Geluk¹,

Tang

2008

J Theor Probab

Self Cite

104

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In this paper we study the asymptotic behavior of the tail probabilities of sums of dependent and real-valued random variables whose distributions are assumed to be subexponential and not necessarily of dominated variation. We propose two general dependence assumptions under which the asymptotic behavior of the tail probabilities of the sums is the same as that in the independent case. In particular, the two dependence assumptions are satisfied by multivariate Farlie-Gumbel-Morgenstern distributions.

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“…The convolution closure of S is from [7]. The other results of parts (ii) and (iii) are from [11] and [10], respectively.…”

Section: Then the Asymptotic Relations In (13) Holdmentioning

confidence: 99%

“…for random variables on the real line; see, e.g. [10], [15], and [22]. For background information on subexponentiality and its applications, the reader is referred to [2], [4], [8], [18].…”

Section: Introductionmentioning

confidence: 99%

Tail behavior of negatively associated heavy-tailed sums

Geluk

2006

Journal of Applied Probability

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Consider a sequence {X k , k ≥ 1} of random variables on (−∞, ∞). Results on the asymptotic tail probabilities of the quantities S n = n k=0 X k , X (n) = max 0≤k≤n X k , and S (n) = max 0≤k≤n S k , with X 0 = 0 and n ≥ 1, are well known in the case where the random variables are independent with a heavy-tailed (subexponential) distribution. In this paper we investigate the validity of these results under more general assumptions. We consider extensions under the assumptions of having long-tailed distributions (the class L) and having the class D ∩ L, where D is the class of distribution functions with dominatedly varying tails. Some results are also given in the case where X k , k ≥ 1, are not necessarily identically distributed and/or independent.

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Asymptotics in the symmetrization inequality

Cited by 6 publications

References 15 publications

Weighted sums of subexponential random variables and asymptotic dependence between returns on reinsurance equities

Weighted sums of subexponential random variables and asymptotic dependence between returns on reinsurance equities

Asymptotic Tail Probabilities of Sums of Dependent Subexponential Random Variables

Tail behavior of negatively associated heavy-tailed sums

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