Hostname: page-component-7dd5485656-npwhs Total loading time: 0 Render date: 2025-10-31T22:51:37.815Z Has data issue: false hasContentIssue false
  • English
  • Français

Aggregation of rapidly varying risks and asymptotic independence

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

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

Information

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.

References

Albrecher, H., Asmussen, S. and Kortschak, D. (2006). Tail asymptotics for the sum of two heavy-tailed dependent risks. Extremes 9, 107–130.Google Scholar
Alink, S., Löwe, M., and Wüthrich, M. V. (2004). Diversification of aggregate dependent risks. Insurance Math. Econom. 35, 77–95.Google Scholar
Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York.Google Scholar
Asmussen, S. and Rojas-Nandayapa, L. (2008). Asymptotics of sums of lognormal random variables with Gaussian copula. Statist. Prob. Lett. 78, 2709–2714.Google Scholar
Davis, R. and Resnick, S. (1988). Extremes of moving averages of random variables from the domain of attraction of the double exponential distribution. Stoch. Process. Appl. 30, 41–68.Google Scholar
De Haan, L. (1970). On Regular Variation and Its Application to the Weak Convergence of Sample Extremes. Mathematisch Centrum, Amsterdam.Google Scholar
De Haan, L. and Ferreira, A. (2006). Extreme Value Theory. Springer, New York.CrossRefGoogle Scholar
Embrechts, P. and Puccetti, G. (2006). Bounds for functions of dependent risks. Finance Stoch. 10, 341–352.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.Google Scholar
Goldie, C. M. and Resnick, S. (1988). Distributions that are both subexponential and in the domain of attraction of an extreme-value distribution. Adv. Appl. Prob. 20, 706–718.CrossRefGoogle Scholar
Klüppelberg, C. and Resnick, S. I. (2008). The Pareto copula, aggregation of risks and the emperor's socks. J. Appl. Prob. 45, 67–84.CrossRefGoogle Scholar
Ko, B. and Tang, Q. (2008). Sums of dependent nonnegative random variables with subexponential tails. J. Appl. Prob. 45, 85–94.Google Scholar
Resnick, S. I. (1971a). Products of distribution functions attracted to extreme value laws. J. Appl. Prob. 8, 781–793.CrossRefGoogle Scholar
Resnick, S. I. (1971b). Tail equivalence and its applications. J. Appl. Prob. 8, 136–156.CrossRefGoogle Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation and Point Processes. Springer, New York.Google Scholar
Rootzén, H. (1986). Extreme value theory for moving average processes. Ann. Prob. 14, 612–652.Google Scholar
Wang, D. and Tang, Q. (2006). Tail probabilities of randomly weighted sums of random variables with dominated variation. Stoch. Models 22, 253–272.Google Scholar