Asymptotic bounds for the distribution of the sum of dependent random variables

Abstract

Suppose that X₁, . . ., 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 P(X₁ + · · · + X_n < s) over all possible dependence structures, denoted by m_n,F(s). We show that m_n,F(ns) → 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 ∈ 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.