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

Part of: Mathematical psychology Distribution theory - Probability

Published online by Cambridge University Press:  30 January 2018

Ruodu Wang
Ruodu Wang*
Affiliation:
University of Waterloo
*
∗ Postal address: Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada. Email address: wang@uwaterloo.ca
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Abstract

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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 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.

Keywords

Dependence boundcomplete mixabilityvalue at riskmodeling uncertainty

MSC classification

Primary: 60E05: Distributions
Secondary: 60E15: Inequalities; stochastic orderings 91E30: Psychophysics and psychophysiology; perception
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 3 , September 2014 , pp. 780 - 798
Copyright
© Applied Probability Trust 

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