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Asymptotic bounds for the distribution of the sum of dependent random variables. (English) Zbl 1320.60045

Summary: Suppose that \(X_{1},\dots, X_{n}\) are random variables with the same known marginal distribution F but unknown dependence structure. In this paper, we study the smallest possible value of \(\mathbb P(X_{1} + \dots + X_{n} < s)\) over all possible dependence structures, denoted by \(m_{n,F}(s)\). We show that \(m_{n,F}(ns) \to 0\) for s no more than the mean of \(F\) under weak assumptions. We also derive a limit of \(m_{n,F}(ns)\) for any \(s \in \mathbb R\) 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.

MSC:

60E05 Probability distributions: general theory
60E15 Inequalities; stochastic orderings
91B30 Risk theory, insurance (MSC2010)
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