Abstract
Suppose thatX1, …,Xnare random variables with the same known marginal distributionFbut unknown dependence structure. In this paper we study the smallest possible value ofP(X1+ · · · +Xn<s) over all possible dependence structures, denoted bymn,F(s). We show thatmn,F(ns) → 0 forsno more than the mean ofFunder weak assumptions. We also derive a limit ofmn,F(ns) for anys∈Rwith an error of at mostn-1/6for 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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