Some results on the optimal matching problem for the Jacobi model

Abstract

We establish some exact asymptotic results for a matching problem with respect to a family of beta distributions. Let $X_1,\dots ,X_n$ be independent random variables with common distribution the symmetric Jacobi measure $d\mu (x)=C_d(1-x^2{)}^{\frac{d}{2}-1}dx$ with dimension $d\ge 1$ on $[-1,1]$, and let ${\mu }_n=\frac{1}{n}{\sum }_{i=1}^n{\delta }_{X_i}$ be the associated empirical measure. We show that $\lim_{n \to \infty} n\E \left[ W_2^2( μ^n, μ) \right] = \sum_{k = 1}^{\infty} \frac{1}{k(k+d-1)}$, where $W_2$ is the quadratic Kantorovich distance with respect to the intrinsic cost $\rho (x,y)=|\arccos (x)-\arccos (y)|$, $(x,y)\in [-1,1{]}^2$, associated to the model. When $\mu$ is the product measure of two Jacobi measures with dimensions $d$ and $d^{\prime }$ respectively, then $\E \left[ W_2^2( μ^n, μ) \right] \approx \frac{\log n}{n}$. In the particular case $d=d^{\prime }=1$ (corresponding to the product of arcsine laws), $\lim_{n \to \infty} \frac{n}{\log n} \E \left[ W_2^2( μ^n, μ) \right] = \fracπ{4}$. Similar results do hold for non-symmetric Jacobi distributions. The proofs are based on the recent PDE and mass transportation approach developed by L.~Ambrosio, F.~Stra and D.~Trevisan.