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Asymptotic Bounds for the Distribution of the Sum of Dependent Random Variables

Part of: Mathematical psychology Distribution theory - Probability

Published online by Cambridge University Press:  30 January 2018

Ruodu Wang
Ruodu Wang*
Affiliation:
University of Waterloo
*
∗ Postal address: Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada. Email address: [email protected]
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Abstract

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Suppose that X1, …, Xn are random variables with the same known marginal distribution F but unknown dependence structure. In this paper we study the smallest possible value of P(X1 + · · · + Xn < s) over all possible dependence structures, denoted by mn,F(s). We show that mn,F(ns) → 0 for s no more than the mean of F under weak assumptions. We also derive a limit of mn,F(ns) for any s ∈ R with an error of at most n-1/6 for general continuous distributions. An application of our result to risk management confirms that the worst-case value at risk is asymptotically equivalent to the worst-case expected shortfall for risk aggregation with dependence uncertainty. In the last part of this paper we present a dual presentation of the theory of complete mixability and give dual proofs of theorems in the literature on this concept.

Keywords

Dependence boundcomplete mixabilityvalue at riskmodeling uncertainty

MSC classification

Primary: 60E05: Distributions
Secondary: 60E15: Inequalities; stochastic orderings 91E30: Psychophysics and psychophysiology; perception
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 3 , September 2014 , pp. 780 - 798
Copyright
© Applied Probability Trust 

References

Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk. Math. Finance 9, 203–228.Google Scholar
Bernard, C., Jiang, X. and Wang, R. (2014). Risk aggregation with dependence uncertainty. Insurance Math. Econom. 54, 93–108.CrossRefGoogle Scholar
Denuit, M., Genest, C. and Marceau, É. (1999). Stochastic bounds on sums of dependent risks. Insurance Math. Econom. 25, 85–104.CrossRefGoogle Scholar
Embrechts, P. and Puccetti, G. (2006). Bounds for functions of dependent risks. Finance Stoch. 10, 341–352.Google Scholar
Embrechts, P. and Puccetti, G. (2010). Risk aggregation. In Copula Theory and Its Applications (Lecture Notes Statist. Proc. 198), Springer, Heidelberg, pp. 111–126.Google Scholar
Embrechts, P., Puccetti, G. and Rüschendorf, L. (2013). Model uncertainty and VaR aggregation. J. Banking Finance 37, 2750–2764.Google Scholar
Gaffke, N. and Rüschendorf, L. (1981). On a class of extremal problems in statistics. Math. Operat. Statist. Ser. Optimization 12, 123–135.Google Scholar
Makarov, G. D. (1982). Estimates for the distribution function of the sum of two random variables when the given marginal distributions are fixed. Theory Prob. Appl. 26, 803–806.CrossRefGoogle Scholar
Puccetti, G. and Rüschendorf, L. (2013). Sharp bounds for sums of dependent risks. J. Appl. Prob. 50, 42–53.CrossRefGoogle Scholar
Puccetti, G. and Rüschendorf, L. (2014). Asymptotic equivalence of conservative value-at-risk- and expected shortfall-based capital charges. J. Risk 16, 1–19.CrossRefGoogle Scholar
Puccetti, G., Wang, B. and Wang, R. (2012). Advances in complete mixability. J. Appl. Prob. 49, 430–440.CrossRefGoogle Scholar
Puccetti, G., Wang, B. and Wang, R. (2013). Complete mixability and asymptotic equivalence of worst-possible VaR and ES estimates. Insurance Math. Econom. 53, 821–828.Google Scholar
Rüschendorf, L. (1982). Random variables with maximum sums. Adv. Appl. Prob. 14, 623–632.Google Scholar
Rüschendorf, L. (2013). Mathematical Risk Analysis. Dependence, Risk Bounds, Optimal Allocations and Portfolios. Springer, Heidelberg.Google Scholar
Rüschendorf, L. and Uckelmann, L. (2002). Variance minimization and random variables with constant sum. In Distributions with Given Marginals and Statistical Modelling. Kluwer, Dordrecht, pp. 211–222.Google Scholar
Wang, R., Peng, L. and Yang, J. (2013). Bounds for the sum of dependent risks and worst value-at-risk with monotone marginal densities. Finance Stoch. 17, 395–417.CrossRefGoogle Scholar
Wang, B. and Wang, R. (2011). The complete mixability and convex minimization problems with monotone marginal densities. J. Multivariate Anal. 102, 1344–1360.CrossRefGoogle Scholar