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Sharp Bounds for Sums of Dependent Risks

Part of: Distribution theory - Probability Mathematical economics

Published online by Cambridge University Press:  30 January 2018

Giovanni Puccetti  and
Ludger Rüschendorf
Giovanni Puccetti*
Affiliation:
University of Firenze
Ludger Rüschendorf*
Affiliation:
University of Freiburg
*
∗ Postal address: Department of Mathematics for Decision Theory, University of Firenze, via delle Pandette, 50127 Firenze, Italy. Email address: giovanni.puccetti@unifi.it
∗∗ Postal address: Department of Mathematical Stochastics, University of Freiburg, Eckerstr. 1, 79104 Freiburg, Germany. Email address: ruschen@stochastik.uni-freiburg.de
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Abstract

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Sharp tail bounds for the sum of d random variables with given marginal distributions and arbitrary dependence structure have been known since Makarov (1981) and Rüschendorf (1982) for d=2 and, in some examples, for d≥3. Based on a duality result, dual bounds have been introduced in Embrechts and Puccetti (2006b). In the homogeneous case, F1=···=Fn, with monotone density, sharp tail bounds were recently found in Wang and Wang (2011). In this paper we establish the sharpness of the dual bounds in the homogeneous case under general conditions which include, in particular, the case of monotone densities and concave densities. We derive the corresponding optimal couplings and also give an effective method to calculate the sharp bounds.

Keywords

Bounds for dependent risksFréchet boundmass transportation theory

MSC classification

Primary: 60E05: Distributions 91B30: Risk theory, insurance
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 1 , March 2013 , pp. 42 - 53
Copyright
Copyright © Applied Probability Trust 2013 

References

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