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Aggregation of rapidly varying risks and asymptotic independence

Part of: Nonparametric inference Limit theorems Distribution theory - Probability Operations research and management science Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Abhimanyu Mitra  and
Sidney I. Resnick
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.

Keywords

RiskGumbelmaximal domain of attractionasymptotic independencesubexponential

MSC classification

Primary: 60E07: Infinitely divisible distributions; stable distributions 60G51: Processes with independent increments; L'evy processes 60G52: Stable processes 60G70: Extreme value theory; extremal processes
Secondary: 60F17: Functional limit theorems; invariance principles 62G30: Order statistics; empirical distribution functions 90B50: Management decision making, including multiple objectives 90C59: Approximation methods and heuristics
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 3 , September 2009 , pp. 797 - 828
Copyright
Copyright © Applied Probability Trust 2009 

Footnotes

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

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