Asymptotic Bounds for the Distribution of the Sum of Dependent Random Variables

Suppose 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 in 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. Key-words: dependence bounds; complete mixability; Value-at-Risk; modeling uncertainty. 2010 Mathematics Subject Classification: 60E05; 60E15; 91E30