The Birkhoff copula scaling-limit conjecture

A Birkhoff copula is a matrix with real entries satisfying the defining conditions of a discrete copula; equivalently, it corresponds to a doubly stochastic matrix. Let ${\mathcal{B}}_n$ be the polytope of Birkhoff copulas, and let ${\mathbf{y}}_n$ denote the associated rescaled fluctuation process. Write $C^0_0({\mathbb{I}}^2)$ for the space of continuous functions on ${\mathbb{I}}^2$ vanishing on the boundary conditions specified for the process, equipped with the supremum norm $\|\cdot\|_{C^0}$. A bridged Brownian sheet is a sample-continuous Brownian sheet conditioned to satisfy the corresponding boundary-bridging conditions. There exist probability spaces $\Omega_n$ with the following properties: on $\Omega_n$ there is a random variable $X_n$ uniformly distributed in ${\mathcal{B}}_n$; on $\Omega_n$ a sample-continuous bridged Brownian sheet process $f$ in $C^0_0({\mathbb{I}}^2)$ is defined; and, for every $\epsilon>0$,

$$\lim_{n\to\infty}\operatorname{Pr}\left[\left\|n\,{\mathbf{y}}_n-f\right\|_{C^0}>\epsilon\right]=0.$$

The conjecture predicts that uniformly sampled Birkhoff copulas have the same Brownian-sheet scaling limit as the discrete-copula model. The paper reports numerical experiments supporting this analogue of the discrete-copula limit theorem; the supplied material does not establish a proof or give a resolution.