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 $X_1, \ldots, X_n$ be independent random variables with common distribution the symmetric Jacobi measure $d\mu (x) = C_d (1-x^2)^{\frac d2 -1} dx$ with dimension $ d \geq 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 $\lim_{n \to \infty} n\E \left[ W_2^2( \mu^n, \mu ) \right] = \sum_{k = 1}^{\infty} \frac{1}{k(k+d-1)}$, 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 $d'$ respectively, then $\E \left[ W_2^2( \mu^n, \mu ) \right] \approx \frac{\log n}{n}$. In the particular case $d = d' = 1$ (corresponding to the product of arcsine laws), $\lim_{n \to \infty} \frac{n}{\log n} \E \left[ W_2^2( \mu^n, \mu ) \right] = \frac{\pi}{4}$. 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.