Journal of Applied Probability

Sharp bounds for sums of dependent risks

Giovanni Puccetti and Ludger Rüschendorf

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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,\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

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

First available in Project Euclid: 20 March 2013

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Zentralblatt MATH identifier

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

Bounds for dependent risks Fréchet bound mass transportation theory


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.

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  • Embrechts, P. and Puccetti, G. (2006a). Aggregating risk capital, with an application to operational risk. Geneva Risk Insurance Rev. 31, 71–90.
  • Embrechts, P. and Puccetti, G. (2006b). Bounds for functions of dependent risks. Finance Stoch.10, 341–352.
  • Gaffke, N. and Rüschendorf, L. (1981). On a class of extremal problems in statistics. Math. Operationsforsch. Statist. Ser. Optimization 12, 123–135.
  • Makarov, G. D. (1981). Estimates for the distribution function of the sum of two random variables with given marginal distributions. Theory Prob. Appl. 26, 803–806.
  • Puccetti, G. and Rüschendorf, L. (2012a). Bounds for joint portfolios of dependent risks. Statist. Risk Modeling 29, 107–132.
  • Puccetti, G. and Rüschendorf, L. (2012b). Computation of sharp bounds on the distribution of a function of dependent risks. J. Comput. Appl. Math. 236, 1833–1840.
  • Puccetti, G., Wang, B. and Wang, R. (2012). Advances in complete mixability. J. Appl. Prob. 49, 430–440.
  • Rüschendorf, L. (1981). Sharpness of Fréchet-bounds. Z. Wahrscheinlichkeitsth. 57, 293–302.
  • Rüschendorf, L. (1982). Random variables with maximum sums. Adv. Appl. Prob. 14, 623–632.
  • Wang, B. and Wang, R. (2011). The complete mixability and convex minimization problems with monotone marginal densities. J. Multivariate Anal. 102, 1344–1360.
  • Wang, R., Peng, L. and Yang, J. (2013). Bounds for the sum of dependent risks and worst Value-at-Risk with monotone marginal densities. To appear in Finance Stoch. \endharvreferences