ASYMPTOTIC BOUNDS FOR THE DISTRIBUTION OF THE SUM OF DEPENDENT RANDOM VARIABLES

作者：Wang Ruodu^*

来源：Journal of Applied Probability, 2014, 51(3): 780-798.

DOI：10.1239/jap/1409932674

• Dependence bound
• complete mixability
• value at risk
• modeling uncertainty

Suppose that X-1, ... , 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(X-1 + ... + X-n %26lt; s) over all possible dependence structures, denoted by m(n,F)(s). We show that m(n,F)(ns) -%26gt; 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 is an element of 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.

• 出版日期2014-9