Journal of Applied Probability

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

Ruodu Wang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Suppose that 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 sR 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.

Article information

J. Appl. Probab., Volume 51, Number 3 (2014), 780-798.

First available in Project Euclid: 5 September 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E05: Distributions: general theory
Secondary: 60E15: Inequalities; stochastic orderings 91E30: Psychophysics and psychophysiology; perception

Dependence bound complete mixability value at risk modeling uncertainty


Wang, Ruodu. Asymptotic bounds for the distribution of the sum of dependent random variables. J. Appl. Probab. 51 (2014), no. 3, 780--798. doi:10.1239/jap/1409932674.

Export citation