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

Abstract

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.

An Academic Response to Basel 3.5

“…, F The asymptotic equivalence of worst VaR and worst ES was established in Puccetti and Rüschendorf [34] in the homogeneous case based on the dual bounds in Embrechts and Puccetti [35] and an assumption of conditional complete mixability. The assumption was later weakened by Puccetti et al [36] and Wang [37] and removed in Wang and Wang [38]. The following extension to inhomogeneous models in the general case was given in (Embrechts et al [39] (Theorem 3.3)).…”

Current open questions in complete mixability

“…Some duality theorems on probability measures with given margins in the literature can be applied to complete and joint mixability. Recent studies on complete mixability using duality methods are found in [29,30,45]. The following theorem was essentially established in [41,37].…”

“…Let F be an arbitrary distribution with bounded support. It has been observed that [e.g 3,45]. when n is large, it is more likely that F becomes n-CM [33].…”

Extremal Dependence Concepts

“…The recent developments of sufficient conditions for joint mixability typically involve techniques in probabilistic combinatorics, used, for instance, in the main results of Wang and Wang (2011), Wang (2012, 2013) and Wang and Wang (2015a). A large class of distributions are asymptotically mixable; see Puccetti, Wang and Wang (2013) and Wang (2014). This property makes joint mixability a flexible concept for the study of high-dimensional problems.…”