Journal of Applied Probability
- J. Appl. Probab.
- Volume 51, Number 3 (2014), 780-798.
Asymptotic bounds for the distribution of the sum of dependent random variables
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.
J. Appl. Probab., Volume 51, Number 3 (2014), 780-798.
First available in Project Euclid: 5 September 2014
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Wang, Ruodu. Asymptotic bounds for the distribution of the sum of dependent random variables. J. Appl. Probab. 51 (2014), no. 3, 780--798. doi:10.1239/jap/1409932674. https://projecteuclid.org/euclid.jap/1409932674