September 2014 Asymptotic bounds for the distribution of the sum of dependent random variables
Ruodu Wang
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J. Appl. Probab. 51(3): 780-798 (September 2014). DOI: 10.1239/jap/1409932674


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 sR 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.


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Ruodu Wang. "Asymptotic bounds for the distribution of the sum of dependent random variables." J. Appl. Probab. 51 (3) 780 - 798, September 2014.


Published: September 2014
First available in Project Euclid: 5 September 2014

zbMATH: 1320.60045
MathSciNet: MR3256227
Digital Object Identifier: 10.1239/jap/1409932674

Primary: 60E05
Secondary: 60E15 , 91E30

Keywords: Complete mixability , Dependence bound , modeling uncertainty , value at risk

Rights: Copyright © 2014 Applied Probability Trust


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Vol.51 • No. 3 • September 2014
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