## Welcome to Ruodu Wang's Homepage Lhasa, China, 2019

## Working Paper Series

Axiomatic theory of risk measures Risk sharing, allocation, and equilibria
Joint mixability Robust risk aggregation
Risk management with risk measures E-values: Theory and methods

# Joint mixability

The series of working papers contains research results on complete mixability (introduced in WP01) and joint mixability (introduced in WP03). The notion of joint mixability, with complete mixability as a special case, leads to the most negative dependence structure among random variables concerning their aggregation, a well-known challenging topic (on the other hand, the most positive dependence, i.e., comonotonicity, is well understood). Joint mixability has wide applications in financial risk management, optimization, and operations research (see the review articles WP08 and WP09). Many results here are useful for the working paper series on Robust risk aggregation. Some basic terminology is listed below.

• A joint mix: a random vector whose components add up to a constant (a center)
• A joint mixable tuple of distributions: the marginals of a joint mix
• An n-completely mixable distribution: the marginal of an n-dimensional joint mix with identical marginals
• An aggregation set: the set of all possible distributions of the sum with specified marginals
• A homogeneous aggregation set: one with identical marginals
• A convex-order lower set: the set of distributions dominated by a specified distribution in convex order
• Some particular challenges and results are:

• Analytical conditions for complete mixability (WP01, WP02, WP04, WP10)
• Analytical conditions for joint mixability (WP03, WP07)
• Analytical structure of an aggregation set (WP06, WP13)
• Connection to other notions of negative dependence (WP09, WP14)
• Numerical procedures and applications (WP03, WP05, WP11, WP12)
• There are many unsolved mathematical questions (WP08). A brief description of each working paper is provided below to explain its main results and the logical structure across papers.

WP14 Joint mixability and negative orthant dependence
(by Takaaki Koike, Liyuan Lin, Ruodu Wang)

This paper connects joint mixability and negative orthant dependence (NOD). A joint mix is not necessarily NOD, but some natural classes of joint mixes are. We find that NOD and the joint mix structure has interesting implications in a multi-marginal optimal transport problem under uncertainty.

WP13 Sums of standard uniform random variables
(by Tiantian Mao, Bin Wang, Ruodu Wang)

This paper contains results on the aggregation set for standard uniform marginals. The main results are a characterization of unimodal densities in this set (Theorem 1) and an identity of the aggregation set and the corresponding convex-order lower set for dimension more than 2 (Theorem 5), thus providing a small partial answer to Open Problem #12 of WP08. The journal version is published in Journal of Applied Probability (2019).

WP12 Combining p-values via averaging
(by Vladimir Vovk, Ruodu Wang)

This paper applies results in this working paper series to multiple hypothesis testing. Main results in WP01, WP03 and WP07 as well as results from Robust risk aggregation are used to determine valid merging methods for p-values without assuming how these p-values are dependent. The journal version is published in Biometrika (2020).

WP11 Dual utilities on risk aggregation under dependence uncertainty
(by Zuoquan Xu, Xunyu Zhou, Ruodu Wang)

The notion of conditional joint mixability is introduced to obtain analytical results on robust risk aggregation for dual utilities (i.e., distortion risk measures). This paper is cross-listed as WP13 of Robust risk aggregation. The journal version is published in Finance and Stochastics (2019).

WP10 Centers of probability measures without the mean
(by Giovanni Puccetti, Pietro Rigo, Bin Wang, Ruodu Wang)

The main result is that the center for n standard Cauchy distributions is not unique and the set of all centers is an interval [-nlog(n-1)/π, nlog(n-1)/π], thus providing an answer to Open Problem #1 of WP08. The journal version is published in Journal of Theoretical Probability (2019).

WP09 Extremal dependence concepts
(by Giovanni Puccetti, Ruodu Wang)

This paper offers a comprehensive review on extremal dependence concepts among random variables, including comonotonicity, countermonotonicity, mutual exclusivity, joint mixability, and a new notion of Σ-countermonotonicitiy. The journal version is published in Statistical Science (2015).

WP08 Current open questions in complete mixability
(by Ruodu Wang)

This survey paper on complete and joint mixability lists 12 open questions. Open Problem #1 is solved in WP10 and Open Problem #12 is partially solved in WP13. The journal version is published in Probability Surveys (2015).

WP07 Joint mixability
(by Bin Wang, Ruodu Wang)

This paper contains some of the most advanced results on joint mixability. In particular it offers a characterization for joint mixability of distributions with decreasing (or increasing) densities (Theorem 3.2), in addition to several results on other distributions and properties of joint mixability. The journal version is published in Mathematics of Operations Research (2016).

WP06 On aggregation sets and lower-convex sets
(by Tiantian Mao, Ruodu Wang)

This paper contains several results on the asymptotic behaviour of the homogenous aggregation set. In particular, the normalized aggregation set converges to a convex-order lower set (Theorem 3.5). The journal version is published in Journal of Multivariate Analysis (2015).

WP05 Detecting complete and joint mixability
(by Giovanni Puccetti and Ruodu Wang)

This paper offers a numerical procedure to check joint mixability based on the Rearrangement Algorithm. The journal version is published in Journal of Computational and Applied Mathematics (2015).

WP04 Complete mixability and asymptotic equivalence of worst-possible VaR and ES estimates
(by Giovanni Puccetti, Bin Wang, Ruodu Wang)

The main results are complete mixability of distributions with a density bounded away from zero (Theorem 3.4) and an asymptotic equivalence formula for VaR/ES (Theorem 4.2, which will be superceded by results in WP06 of Robust risk aggregation). The journal version is published in Insurance: Mathematics and Economics (2013).

WP03 Bounds for the sum of dependent risks and worst Value-at-Risk with monotone marginal densities (by Ruodu Wang, Liang Peng, Jingping Yang)

In this paper, the term joint mixability is coined. The main results are sharp bounds on the distribution function and VaR of aggregate risk with homogeneous marginals and monotone densities (Theorem 3.6 and Corollary 3.7). This paper is cross-listed as WP01 of Robust risk aggregation. The journal version is published in Finance and Stochastics (2013).

WP02 Advances in complete mixability
(by Giovanni Puccetti, Bin Wang, Ruodu Wang)

This paper contains several fundamental properties of complete mixability (Theorems 3.1 and 3.2) and a proof of the complete mixability of concave densities (Theorem 4.2). The journal version is published in Journal of Applied Probability (2012).

WP01 The complete mixability and convex minimization problems for monotone marginal densities (by Bin Wang, Ruodu Wang)

In this paper, the term complete mixability is coined. There are two main results. First, a distribution with monotone density is completely mixable if and only if the mean condition is satisfied (Theorem 2.4 and Corollary 2.9). Second, the smallest convex-order element of a homogeneous aggregation set with monotone densities is obtained (Theorem 3.4). The journal version is published in Journal of Multivariate Analysis (2011).