Journal of Applied Probability

Sharp bounds for sums of dependent risks

Giovanni Puccetti and Ludger Rüschendorf

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Abstract

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,\break $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.

Article information

Source
J. Appl. Probab., Volume 50, Number 1 (2013), 42-53.

Dates
First available in Project Euclid: 20 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.jap/1363784423

Digital Object Identifier
doi:10.1239/jap/1363784423

Mathematical Reviews number (MathSciNet)
MR3076771

Zentralblatt MATH identifier
1282.60017

Subjects
Primary: 60E05: Distributions: general theory 91B30: Risk theory, insurance

Keywords
Bounds for dependent risks Fréchet bound mass transportation theory

Citation

Puccetti, Giovanni; Rüschendorf, Ludger. Sharp bounds for sums of dependent risks. J. Appl. Probab. 50 (2013), no. 1, 42--53. doi:10.1239/jap/1363784423. https://projecteuclid.org/euclid.jap/1363784423


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References

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