Scaling limits of discrete copulas are bridged Brownian sheets

Abstract

For large $n$, take a random $n\times n$ permutation matrix and its associated discrete copula $X_n$. For $a,b=0,1,\dots ,n$, let $y_n(\frac{a}{n},\frac{b}{n})=\frac{1}{n}(X_{a,b}-\frac{ab}{n})$; define $y_n:[0,1{]}^2\to R$ by interpolating quadratically on squares of side $\frac{1}{n}$. We prove a Donsker type central limit theorem: $\sqrt{n}{y}_n$ approaches a bridged Brownian sheet on the unit square.

29 pages, 5 figures