# Scaling limits of discrete copulas are bridged Brownian sheets

Research paper by **Juliana Freire, Nicolau C. Saldanha, Carlos Tomei**

Indexed on: **13 Jan '16**Published on: **13 Jan '16**Published in: **Mathematics - Probability**

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#### Abstract

For large $n$, take a random $n \times n$ permutation matrix and its
associated discrete copula $X_n$. For $a, b = 0, 1, \ldots, n$, 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.