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Aggregation of rapidly varying risks and asymptotic independence

Published online by Cambridge University Press:  01 July 2016

Abhimanyu Mitra*
Affiliation:
Cornell University
Sidney I. Resnick*
Affiliation:
Cornell University
*
Postal address: School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853, USA.
Postal address: School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853, USA.
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Abstract

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We study the tail behavior of the distribution of the sum of asymptotically independent risks whose marginal distributions belong to the maximal domain of attraction of the Gumbel distribution. We impose conditions on the distribution of the risks (X, Y) such that P(X + Y > x) ∼ (constant) P(X > x). With the further assumption of nonnegativity of the risks, the result is extended to more than two risks. We note a sufficient condition for a distribution to belong to both the maximal domain of attraction of the Gumbel distribution and the subexponential class. We provide examples of distributions which satisfy our assumptions. The examples include cases where the marginal distributions of X and Y are subexponential and also cases where they are not. In addition, the asymptotic behavior of linear combinations of such risks with positive coefficients is explored, leading to an approximate solution of an optimization problem which is applied to portfolio design.

Type
General Applied Probability
Copyright
Copyright © Applied Probability Trust 2009 

Footnotes

Research partially supported by ARO contract W911NF-07-1-0078 at Cornell University.

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