Copula Realizations and Centers of Complete Mixes

Definitions and mathematical statement

Under the usual real-valued interpretation of the source, let $n$ be a positive integer and let $F_1,\ldots,F_n$ be distribution functions on ${\mathbb{R}}$. A coupling of these margins is a random vector $(X_1,\ldots,X_n)$ with $X_i\sim F_i$ for every $i$. It is a joint mix if, for some $k\in{\mathbb{R}}$, \[ \sum_{i=1}^{n}X_i=k\qquad\text{almost surely}. \] Write \[ \mathcal S(F_1,\ldots,F_n)=\left\{k\in{\mathbb{R}}:\text{there exists such a coupling with }\sum_{i=1}^{n}X_i=k\text{ almost surely}\right\}. \] A distribution function $F$ is $n$-completely mixable if $\mathcal S(F,\ldots,F)\ne\varnothing$. To distinguish possible conventions for the word "center," call $k$ the constant sum and $k/n$ its normalized center; the supplied record does not specify which convention it uses.

An $n$-copula is the joint distribution function of a random vector $(U_1,\ldots,U_n)$ whose individual coordinates are uniform on $[0,1]$. For $0<u<1$, define the generalized inverse \[ Q_F(u)=\inf\{x\in{\mathbb{R}}:F(x)\ge u\}. \] Values at the endpoints do not affect the following almost-sure condition.

Copula realization question. Given an $n$-completely mixable distribution $F$, describe a copula $C$ and a constant $k\in\mathcal S(F,\ldots,F)$ such that a random vector with copula $C$ and uniform margins satisfies \[ \sum_{i=1}^{n}Q_F(U_i)=k\qquad\text{almost surely}. \] Then $(Q_F(U_1),\ldots,Q_F(U_n))$ realizes an $n$-complete mix with every margin equal to $F$. The source does not specify a prescribed center, an explicitness criterion, or a restricted class of copulas. Complete mixability already includes existence of a suitable coupling; the additional construction or description intended by the question is unspecified.

Center question, with unresolved hypotheses. The source asks whether uniqueness of the center of a joint mix can be proved by a more elegant analytic argument. For prescribed margins, the substantive uniqueness issue concerns whether $\mathcal S(F_1,\ldots,F_n)$ contains at most one element, rather than the constant attached to a fixed coupling. The record does not state the hypotheses of the intended uniqueness result, identify the existing argument to be improved, or specify whether this question concerns general margins or only identical margins. No moment, support, or other restriction is supplied. Consequently, an unconditional uniqueness assertion cannot be extracted from this record; the request for an analytic proof remains qualified by these missing specifications.

Short English statement

For an $n$-completely mixable distribution, describe a copula coupling identical margins so that their sum is constant. The record also requests a more elegant analytic proof of center uniqueness, but does not specify the relevant hypotheses or existing proof. These omissions prevent a precise unconditional uniqueness statement.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30002858$, source problem code catalogue entry, title "Copula Realizations and Centers of Complete Mixes," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2015.
• Supplied report citation: Copulae: On the Crossroads of Mathematics and Economics (2016), Oberwolfach Reports, pp. 1085--1137, DOI 10.4171/owr/2015/20. Source URL and literature entry "Oberwolfach Report 20": https://doi.org/10.4171/owr/2015/20. The background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature reference: Wang, Current Open Questions in Complete Mixability, https://doi.org/10.1214/14-PS250.
• Supplied literature reference: Komisarski--Labuschagne, https://doi.org/10.1007/s10959-023-01277-2. No title or more detailed bibliographic information is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the record open and reports partial structural and nonuniqueness results in later work, while stating that the requested copula construction and the general uniqueness/analytic-proof issues are not settled together. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.
• The registry assigns statement_status: corrected_verified and says that mathematical content was preserved while notation, typography, or omitted definitions were repaired for publication. These are supplied editorial labels, not mathematical verification of this draft.